In [Er73] and [ErGr80] the generalisation where $A\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.)
See also [350].
An alternative, simpler, proof was given by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22], who improved the bound on the smallest modulus to $616000$.
Even the case $k=3$ is non-trivial, but was proved by Bloom and Sisask [BlSi20]. Much better bounds for $r_3(N)$ were subsequently proved by Kelley and Meka [KeMe23]. Green and Tao [GrTa17] proved $r_4(N)\ll N/(\log N)^{c}$ for some small constant $c>0$. Gowers [Go01] proved \[r_k(N) \ll \frac{N}{(\log\log N)^{c_k}},\] where $c_k>0$ is a small constant depending on $k$. The current best bounds for general $k$ are due to Leng, Sah, and Sawhney [LSS24], who show that \[r_k(N) \ll \frac{N}{\exp(-(\log\log N)^{c_k})}\] for some constant $c_k>0$ depending on $k$.
Curiously, Erdős [Er83c] thought this conjecture was the 'only way to approach' the conjecture that there are arbitrarily long arithmetic progressions of prime numbers, now a theorem due to Green and Tao [GrTa08].
Indeed, the answer is yes, as proved by Banks, Freiberg, and Turnage-Butterbaugh [BFT15] with an application of the Maynard-Tao machinery concerning bounded gaps between primes [Ma15]. They in fact prove that, for any $m\geq 1$, there are infinitely many $n$ such that \[d_n<d_{n+1}<\cdots <d_{n+m}\] and infinitely many $n$ such that \[d_n> d_{n+1}>\cdots >d_{n+m}.\]
Hough and Nielsen [HoNi19] proved that at least one modulus must be divisible by either $2$ or $3$. A simpler proof of this fact was provided by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [BBMST22].
Selfridge has shown (as reported in [Sc67]) that such a covering system exists if a covering system exists with moduli $n_1,\ldots,n_k$ such that no $n_i$ divides any other $n_j$ (but the latter has been shown not to exist, see [586]).
Granville and Soundararajan [GrSo98] have conjectured that at most $3$ powers of 2 suffice for all odd integers, and hence at most $4$ powers of $2$ suffice for all even integers. (The restriction to odd integers is important here - for example, Bogdan Grechuk has observed that $1117175146$ is not the sum of a prime and at most $3$ powers of $2$, and pointed out that parity considerations, coupled with the fact that there are many integers not the sum of a prime and $2$ powers of $2$ (see [9]) suggest that there exist infinitely many even integers which are not the sum of a prime and at most $3$ powers of $2$).
This is equivalent to asking whether every $n$ not divisible by $4$ is the sum of a squarefree number and a power of two. Erdős thought that proving this with two powers of 2 is perhaps easy, and could prove that it is true (with a single power of two) for almost all $n$.
Tao [Ta23] has proved that this series does converge assuming a strong form of the Hardy-Littlewood prime tuples conjecture.
Are there infinitely many practical $m$ such that \[h(m) < (\log\log m)^{O(1)}?\] Is it true that $h(n!)<n^{o(1)}$?
The answer is no, as proved by Filaseta, Ford, Konyagin, Pomerance, and Yu [FFKPY07], who (among other results) prove that if \[1< C \leq N^{\frac{\log\log\log N}{4\log\log N}}\] then, for any $N\leq n_1<\cdots< n_k\leq CN$, the density of integers not covered for any fixed choice of residue classes is at least \[\prod_{i}(1-1/n_i)\] (and this density is achieved for some choice of residue classes as above).
Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.
Erdős and Sárközy conjectured the stronger version that if $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.
See also [40].
This is extremely false, as shown by Konieczny [Ko15], who both constructs an explicit permutation with $S(\pi) \geq n^2/4$, and also shows that for a random permutation we have \[S(\pi)\sim \frac{1+e^{-2}}{4}n^2.\]
See also [356].
For all sufficiently large $N$, if $A\sqcup B=\{1,\ldots,2N\}$ is a partition into two equal parts, so that $\lvert A\rvert=\lvert B\rvert=N$, then there is some $x$ such that the number of solutions to $a-b=x$ with $a\in A$ and $b\in B$ is at least $cN$.
Can a lacunary set $A\subset\mathbb{N}$ be an essential component?
Solved by Tao [Ta23b], who proved that \[ \lvert A\rvert \leq \left(1+O\left(\frac{(\log\log x)^5}{\log x}\right)\right)\pi(x).\]
There is likely nothing special about the integers in this question, and indeed Erdős and Szemerédi also ask a similar question about finite sets of real or complex numbers. The current best bound for sets of reals is the same bound of Rudnev and Stevens above. The best bound for complex numbers is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{5}{4}},\] due to Solymosi [So05].
One can in general ask this question in any setting where addition and multiplication are defined (once one avoids any trivial obstructions such as zero divisors or finite subfields). For example, it makes sense for subsets of finite fields. The current record is that if $A\subseteq \mathbb{F}_p$ with $\lvert A\rvert <p^{5/8}$ then \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{11}{9}+o(1)},\] due to Rudnev, Shakan, and Shkredov [RSS20].
There is also a natural generalisation to higher-fold sum and product sets. For example, it is reasonable to conjecture (although I cannot find a record of Erdős himself actually doing so) that for any $m\geq 2$ and finite set of integers (or reals, etc) $A$ that \[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\] See [53] for more on this generalisation.
Burr and Erdős [BuEr85] showed that there exists a constant $c>0$ such that it cannot be true that \[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\] for all large $N$ and that there exists a Ramsey $2$-complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert < (2\log_2N)^3.\] Improve either of these bounds.
Solved by Conlon, Fox, and Pham [CFP21], who constructed for every $r\geq 2$ an $r$-Ramsey complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,\] and showed that this is best possible, in that there exists some constant $c>0$ such that if $A\subset \mathbb{N}$ satisfies \[\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2\] for all large $N$ then $A$ cannot be $r$-Ramsey complete.
See also [54].
Erdős later asked [Er95] if the conjecture remains true provided $N\geq (1+o(1))p_k^2$ (or, in a weaker form, whether it is true for $N$ sufficiently large depending on $k$).
A stronger form was established by Gao, Huo, and Ma [GaHuMa21], who proved that if a graph $G$ has chromatic number $\chi(G)\geq 2k+3$ then $G$ contains cycles of $k+1$ consecutive odd lengths.
He, Ma, and Yang [HeMaYa21] have proved this conjecture when $n=q^2+q+1$ for some even integer $q$.
This was solved in the affirmative if the minimum degree is larger than some absolute constant by Liu and Montgomery [LiMo20] (therefore disproving the above stronger conjecture of Erdős and Gyárfás). Liu and Montgomery prove a much stronger result: if the average degree of $G$ is sufficiently large then there is some large integer $\ell$ such that for every even integer $m\in [(\log \ell)^8,\ell]$, $G$ contains a cycle of length $m$.
Solved by Verstraëte [Ve05], who gave a non-constructive proof that such a set $A$ exists.
Liu and Montgomery [LiMo20] proved that in fact this is true when $A$ is the set of powers of $2$ (more generally any set of even numbers which doesn't grow too quickly) - in particular this contradicts the previous belief of Erdős.
The answer is yes, proved by Gruslys and Letzter [GrLe20].
Are there infinitely many graphs $G$ which are not Ramsey size linear but such that all of its subgraphs are?
See also [600] and the entry in the graphs problem collection.
A split graph is one where the vertices can be split into a clique and an independent set. Every split graph is chordal. Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\frac{3}{16}n^2+O(n)$ many cliques.
Proved by Ahlswede and Khachatrian [AhKh97], who more generally showed the following. Let $2\leq t\leq k\leq m$ and let $r\geq 0$ be such that \[\frac{1}{r+1}\leq \frac{m-2k+2t-2}{(t-1)(k-t+1)}< \frac{1}{r}.\] The largest possible family of subsets of $[m]$ of size $k$, such that the pairwise intersections have size at least $t$, is the family of all subsets of $[m]$ of size $k$ which contain at least $t+r$ elements from $\{1,\ldots,t+2r\}$.
A similar question can be asked for other even cycles.
Even stronger, is there some $c>0$ such that, for all large $k$, $R(G)>cR(k)$ for every graph $G$ with chromatic number $\chi(G)=k$?
Since $R(k)\leq 4^k$ this is trivial for $\epsilon\geq 3/4$. Yuval Wigderson points out that $R(G)\gg 2^{k/2}$ for any $G$ with chromatic number $k$ (via a random colouring), which asymptotically matches the best-known lower bounds for $R(k)$.
See also the entry in the graphs problem collection and this second one.
A stronger form (see [604]) may be true: is there a single point which determines $\gg n/\sqrt{\log n}$ distinct distances, or even $\gg n$ many such points, or even that this is true averaged over all points.
Part of the difficulty of this problem is explained by a result of Valtr (see [Sz16]), who constructed a metric on $\mathbb{R}^2$ and a set of $n$ points with $\gg n^{4/3}$ unit distance pairs (with respect to this metric). The methods of the upper bound proof of Spencer, Szemerédi, and Trotter [SST84] generalise to include this metric. Therefore to prove an upper bound better than $n^{4/3}$ some special feature of the Euclidean metric must be exploited.
See a survey by Szemerédi [Sz16] for further background and related results.
Edelsbrunner and Hajnal [EdHa91] have constructed $n$ such points with $2n-7$ pairs distance $1$ apart. (This disproved an early stronger conjecture of Erdős and Moser, that the true answer was $\frac{5}{3}n+O(1)$.)
If this fails for $4$, perhaps there is some constant for which it holds?
Erdős suggested this as an approach to solve [96]. Indeed, if this problem holds for $k+1$ vertices then, by induction, this implies an upper bound of $kn$ for [96].
The answer is no if we omit the requirement that the polygon is convex (I thank Boris Alexeev and Dustin Mixon for pointing this out), since for any $d$ there are graphs with minimum degree $d$ which can be embedded in the plane such that each edge has length one (for example one can take the $d$-dimensional hypercube graph on $2^d$ vertices). One can then connect the vertices in a cyclic order so that there are no self-intersections and no three consecutive vertices on a line, thus forming a (non-convex) polygon.
Grünbaum [Gr76] constructed an example with $\gg n^{3/2}$ such lines. Erdős speculated this may be the correct order of magnitude. This is false: Solymosi and Stojaković [SoSt13] have constructed a set with no five on a line and at least \[n^{2-O(1/\sqrt{\log n})}\] many lines containing exactly four points.
See also [102]. A generalisation of this problem is asked in [588].
It is easy to see that $h_c(n) \ll_c n^{1/2}$, and Erdős originally suggested that perhaps a similar lower bound $h_c(n)\gg_c n^{1/2}$ holds. Zach Hunter has pointed out that this is false, even replacing $>3$ points on each line with $>k$ points: consider the set of points in $\{1,\ldots,m\}^d$ where $n\approx m^d$. These intersect any line in $\ll_d n^{1/d}$ points, and have $\gg_d n^2$ many pairs of points each of which determine a line with at least $k$ points. This is a construction in $\mathbb{R}^d$, but a random projection into $\mathbb{R}^2$ preserves the relevant properties.
This construction shows that $h_c(n) \ll n^{1/\log(1/c)}$.
See also [146] and [147] and the entry in the graphs problem collection.
The Chebyshev polynomials show that $n^2/2$ is best possible here. Erdős originally conjectured this without the $o(1)$ term but Szabados observed that was too strong. Pommerenke [Po59a] proved an upper bound of $\frac{e}{2}n^2$.
Eremenko and Lempert [ErLe94] have shown this is true, and in fact Chebyshev polynomials are the extreme examples.
Wagner [Wa88] proves, for $n\geq 3$, the existence of such polynomials with \[\mu(A) \ll_\epsilon (\log\log n)^{-1/2+\epsilon}\] for all $\epsilon>0$.
For example, Larson [La00] has shown that this is false when $\alpha=\omega^{\omega^2}$ and $n=5$. There is more background and proof sketches in Chapter 2.9 of [HST10], by Hajnal and Larson.
This was solved by Beck [Be91], who proved that there exists some $c>0$ such that \[\max_{n\leq N} M_n > N^c.\]
This is true if $A$ is unbounded or dense in some interval. It therefore suffices to prove this when $A=\{a_1>a_2>\cdots\}$ is a countable strictly monotone sequence which converges to $0$.
Steinhaus [St20] has proved this is false whenever $A$ is a finite set.
This conjecture is known in many special cases (but, for example, it is is open when $A=\{1,1/2,1/4,\ldots\}$. For an overview of progress we recommend a nice survey by Svetic [Sv00] on this problem.
Keevash and Sudakov [KeSu06] have proved this under the additional assumption that $G$ has at most $n^2/12$ edges.
See also [3].
For fixed $n,t$ as we change $\alpha$ from $0$ to $1/2$ does $F^{(t)}(n,\alpha)$ increase continuously or are there jumps? Only one jump?
Erdős believed there might be just one jump, occcurring at $\alpha=0$.
See also [563].
Prove that for every fixed $0\leq \alpha \leq 1/2$, as $n\to\infty$, \[F(n,\alpha)\sim c_\alpha \log n\] for some constant $c_\alpha$.
See also [544].
Is \[\lim_{k\to \infty}\frac{f(k)}{\log W(k)}=\infty\] where $W(k)$ is the van der Waerden number?
Moreira [Mo17] has proved that in any finite colouring of $\mathbb{N}$ there exist $x,y$ such that $\{x,x+y,xy\}$ are all the same colour.
Alweiss [Al23] has proved that, in any finite colouring of $\mathbb{Q}\backslash \{0\}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour. Bowen and Sabok [BoSa22] had proved this earlier for the first non-trivial case of $\lvert A\rvert=2$.
Sets known to be Ramsey include vertices of $k$-dimensional rectangles [EGMRSS73], non-degenerate simplices [FrRo90], trapezoids [Kr92], and regular polygons/polyhedra [Kr91].
More generally, if $f(n)$ is the largest integer such that, for some prime $p$, we have $p^{f(n)}$ dividing $\binom{2n}{n}$, then $f(n)$ should tend to infinity with $n$. Can one even disprove that $f(n)\gg \log n$?
Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]
This is trivially true if $\mathcal{F}$ does not contain any bipartite graphs, since by the Erdős-Stone theorem if $H\in\mathcal{F}$ has minimal chromatic number $r\geq 2$ then \[\mathrm{ex}(n;H)=\mathrm{ex}(n;\mathcal{F})=\left(\frac{r-2}{r-1}+o(1)\right)\binom{n}{2}.\] Erdős and Simonovits observe that this is false for infinite families $\mathcal{F}$, e.g. the family of all cycles.
See also [575] and the entry in the graphs problem collection.
A construction due to Pyber, Rödl, and Szemerédi [PRS95] shows that this is best possible.
See also [483].
The answer is yes, which is a corollary of the density Hales-Jewett theorem, proved by Furstenberg and Katznelson [FuKa91].
This is false; Kovač [Ko23] provides an explicit (and elegantly simple) colouring using 25 colours such that no colour class contains the vertices of a rectangle of area $1$. The question for parallelograms remains open.
In the same article Rödl also proved a lower bound for this problem, constructing, for all $n$, a $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ such that if $X\subseteq \{2,\ldots,n\}$ is such that $\binom{X}{2}$ is monochromatic then \[\sum_{x\in X}\frac{1}{\log x}\ll \log\log\log n.\]
This bound is best possible, as proved by Conlon, Fox, and Sudakov [CFS13], who proved that, if $n$ is sufficiently large, then in any $2$-colouring of $\binom{\{2,\ldots,n\}}{2}$ there exists some $X\subset \{2,\ldots,n\}$ such that $\binom{X}{2}$ is monochromatic and \[\sum_{x\in X}\frac{1}{\log x}\geq 2^{-8}\log\log\log n.\]
This problem is equivalent to one on 'abelian squares' (see [231]). In particular $A$ can be interpreted as an infinite string over an alphabet with $d$ letters (each letter describining which of the $d$ possible steps is taken at each point). An abelian square in a string $s$ is a pair of consecutive blocks $x$ and $y$ appearing in $s$ such that $y$ is a permutation of $x$. The connection comes from the observation that $p,q,r\in A\subset \mathbb{R}^d$ form a three-term arithmetic progression if and only if the string corresponding to the steps from $p$ to $q$ is a permutation of the string corresponding to the steps from $q$ to $r$.
This problem is therefore equivalent to asking for which $d$ there exists an infinite string over $\{1,\ldots,d\}$ with no abelian squares. It is easy to check that in fact any finite string of length $7$ over $\{1,2,3\}$ contains an abelian square.
An infinite string without abelian squares was constructed when $d=4$ by Keränen [Ke92]. We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results, and a blog post by Renan for an entertaining and educational discussion.
Is it true that, for almost all $x$, for sufficiently large $n$, we have \[R_{n+1}(x)=R_n(x)+\frac{1}{m},\] where $m$ is minimal such that $m$ does not appear in $R_n(x)$ and the right-hand side is $<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?)
Without the 'eventually' condition this can fail for some rational $x$ (although Erdős [Er50b] showed it holds without the eventually for rationals of the form $1/m$). For example \[R_1(\tfrac{11}{24})=\frac{1}{3}\] but \[R_2(\tfrac{11}{24})=\frac{1}{4}+\frac{1}{5}.\]
See also [489].
The Sylvester-Gallai theorem implies that there must exist a point where only two lines from $A$ meet. This problem asks whether there must exist three such points which form a triangle (with sides induced by lines from $A$). Füredi and Palásti [FuPa84] showed this is false when $d\geq 4$ is not divisible by $9$. Escudero [Es16] showed this is false for all $d\geq 4$.
That $f(n)\to \infty$ was proved by Motzkin [Mo51]. Kelly and Moser [KeMo58] proved that $f(n)\geq\tfrac{3}{7}n$ for all $n$. This is best possible for $n=7$. Motzkin conjectured that for $n\geq 13$ there are at least $n/2$ such lines. Csima and Sawyer [CsSa93] proved a lower bound of $f(n)\geq \tfrac{6}{13}n$ when $n\geq 8$. Green and Tao [GrTa13] proved that $f(n)\geq n/2$ for sufficiently large $n$. (A proof that $f(n)\geq n/2$ for large $n$ was earlier claimed by Hansen but this proof was flawed.)
The bound of $n/2$ is best possible for even $n$, since one could take $n/2$ points on a circle and $n/2$ points at infinity. Surprisingly, Green and Tao [GrTa13] show that if $n$ is odd then $f(n)\geq 3\lfloor n/4\rfloor$.
In [Er84] Erdős speculates that perhaps there are $\geq (1+o(1))kn/6$ many such lines, but says 'perhaps [this] is too optimistic and one should first look for a counterexample'. The constant $1/6$ would be best possible here, since there are arrangements of $n$ points with no four points on a line and $\sim n^2/6$ many lines containing three points (see Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84]).
Indeed, Shaffaf and Tao actually proved that such a rational distance set must be contained in a finite union of real algebraic curves. Solymosi and de Zeeuw [SdZ10] then proved (unconditionally) that a rational distance set contained in a real algebraic curve must be finite, unless the curve contains a line or a circle.
Ascher, Braune, and Turchet [ABT20] observed that, combined, these facts imply that a rational distance set in general position must be finite (conditional on the Bombieri-Lang conjecture).
Ascher, Braune, and Turchet [ABT20] have shown that there is a uniform upper bound on the size of such a set, conditional on the Bombieri-Lang conjecture. Greenfeld, Iliopoulou, and Peluse [GIP24] have shown (unconditionally) that any such set must be very sparse, in that if $S\subseteq [-N,N]^2$ has no three on a line and no four on a circle, and all pairwise distances integers, then \[\lvert S\rvert \ll (\log N)^{O(1)}.\]
In fact, such a set does exist, as proved by Jackson and Mauldin [JaMa02]. Their construction depends on the axiom of choice.
See also [230].
For more details see the paper [BoBo09] of Bombieri and Bourgain and where Kahane's construction is improved to yield such a polynomial with \[P(z)=\sqrt{n}+O(n^{\frac{7}{18}}(\log n)^{O(1)})\] for all $z\in\mathbb{C}$ with $\lvert z\rvert=1$.
See also [228].
Erdős then asked if there is in fact an infinite string formed from $\{1,2,3,4\}$ which contains no abelian squares? This is equivalent to [192], and such a string was constructed by Keränen [Ke92]. The existence of this infinite string gives a negative answer to the problem for all $k\geq 4$.
Containing no abelian squares is a stronger property than being squarefree (the existence of infinitely long squarefree strings over alphabets with $k\geq 3$ characters was established by Thue).
We refer to a recent survey by Fici and Puzynina [FiPu23] for more background and related results.
Estimate \[m_1=\sup \overline{\delta}(A),\] where $A$ ranges over all measurable subsets of $\mathbb{R}^2$ without two points distance $1$ apart. In particular, is $m_1\leq 1/4$?
The trivial upper bound is $m_1\leq 1/2$, since for any unit vector $u$ the sets $A$ and $A+u$ must be disjoint. Erdős' question was solved by Ambrus, Csiszárik, Matolcsi, Varga, and Zsámboki [ACMVZ23] who proved that $m_1\leq 0.247$.
Schinzel conjectured the generalisation that, for any fixed $a$, if $n$ is sufficiently large in terms of $a$ then there exist distinct integers $1\leq x<y<z$ such that \[\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.\]
The answer is yes, proved by Freiman [Fr73].
Essentially the best possible result was proved by Tijdeman and Wagner [TiWa80], who proved that, for almost all intervals of the shape $[0,x)$, we have \[\limsup_{N\to \infty}\frac{\lvert D_N([0,x))\rvert}{\log N}\gg 1.\]
If we drop the non-empty requirement then Simonovits, Sós, and Graham [SiSoGr80] have shown that \[t\leq \binom{N}{3}+\binom{N}{2}+\binom{N}{1}+1\] and this is best possible.
For $k=1$ or $k=2$ any set $A$ such that $\sum_{n\in A}\frac{1}{n}=\infty$ has this property.
Is it true that for every $\epsilon>0$ there exists some $k$ such that the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?
Does this process always terminate if $x$ has odd denominator and $A$ is the set of odd numbers? More generally, for which pairs $x$ and $A$ does this process terminate?
Graham [Gr64b] has shown that $\frac{m}{n}$ is the sum of distinct unit fractions with denominators $\equiv a\pmod{d}$ if and only if \[\left(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\right)=1.\] Does the greedy algorithm always terminate in such cases?
Graham [Gr64c] has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always.
This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large $N$, there exists a prime $p\in [N,2N]$ such that $\frac{p+1}{2}$ is also prime.
An elementary inductive argument shows that $n_k\leq ku_k$ where $u_1=1$ and $u_{i+1}=u_i(u_i+1)$, and hence \[v(k) \leq kc_0^{2^k},\] where \[c_0=\lim_n u_n^{1/2^n}=1.26408\cdots\] is the 'Vardi constant'.
Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] (coupled with the trivial greedy approach) implies that $k(N)=(1-o(1))\log N$.
Steinerberger [St24] has shown that in fact the number of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in A}\frac{1}{n}\leq 1$ is at most $2^{0.93N}$.
Independently, Liu and Sawhney [LiSa24] gave a best possible answer, proving that the number of $A\subseteq \{1,\ldots,N\}$ with $\sum_{n\in A}\frac{1}{n}=1$ is \[2^{(0.91\cdots+o(1))N},\] where $0.91\cdots$ is an explicit number defined as the solution to a certain integral equation. The same asymptotic was proved, also independently (and with a different proof) by Conlon, Fox, He, Mubayi, Pham, Suk, and Verstraëte [CFHMPSV24], who prove more generally, for any $x\in \mathbb{Q}_{>0}$, a similar expression for the number of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in A}\frac{1}{n}=x$.
Liu and Sawhney [LiSa24] have proved that $A(N)=(1-1/e+o(1))N$.
The possible alternative question, that if $A\subseteq \mathbb{N}$ is a set of positive lower density then must there exist $a,b,c\in A$ such that \[\frac{1}{a}=\frac{1}{b}+\frac{1}{c},\] has a negative answer, taking for example $A$ to be the union of $[5^k,(1+\epsilon)5^k]$ for large $k$ and sufficiently small $\epsilon>0$. This was observed by Hunter and Sawhney.
Related to [18].
This was solved by Yokota [Yo88], who proved that \[D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2.\] This was improved by Liu and Sawhney [LiSa24] to \[D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}.\]
In [ErGr80] this problem is stated with the sequence $u_1=1$ and $u_{n+1}=u_n(u_n+1)$, but Quanyu Tang has pointed out this is probably an error (since with that choice we do not have $\sum \frac{1}{u_n}=1$). This question with Sylvester's sequence is the most natural interpretation of what they meant to ask.
This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample. More generally, Sándor shows that for any $n\geq 2$ there exists a finite set $A\subseteq \mathbb{N}\backslash\{1\}$ with $\sum_{k\in A}\frac{1}{k}<n$ and no partition into $n$ parts each of which has $\sum_{k\in A_i}\frac{1}{k}<1$.
The minimal counterexample is $\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart.
See also [321].
See also [320].
Independently Erdős [Er36] and Chowla proved that for all $k\geq 3$ and infinitely many $n$ \[1_A^{(k)}(n) \gg n^{c/\log\log n}\] for some constant $c>0$ (depending on $k$).
For $k>2$ it is not known if $f_{k,k}(x)=o(x)$.
What if $a,b\in A$ with $a\neq b$ implies $a+b\nmid 2ab$? Must $\lvert A\rvert=o(N)$?
One can also ask what conditions are sufficient for $D(A)$ to have positive density, or for $\sum_{d\in D(A)}\frac{1}{d}=\infty$, or even just $D(A)\neq\emptyset$.
It is likely that $f(n)\leq n^{o(1)}$, or even $f(n)\leq e^{O(\sqrt{\log n})}$.
The set of squares has order $4$ and restricted order $5$ (see [Pa33]) and the set of triangular numbers has order $3$ and restricted order $3$ (see [Sc54]).
Is it true that if $A\backslash F$ is a basis for all finite sets $F$ then $A$ must have a restricted order? What if they are all bases of the same order?
This sequence is at OEIS A005282.
The original question was answered by Szemerédi and Vu [SzVu06] (who proved that the answer is yes).
This is best possible, since Folkman [Fo66] showed that for all $\epsilon>0$ there exists a multiset $A$ with \[\lvert A\cap \{1,\ldots,N\}\rvert\ll N^{1+\epsilon}\] for all $N$, such that $A$ is not subcomplete.
This is true, and was proved by Szemerédi and Vu [SzVu06]. The stronger conjecture that this is true under \[\lvert A\cap \{1,\ldots,N\}\rvert\geq (2N)^{1/2}\] seems to be still open (this would be best possible as shown by [Er61b].
Is it true that there are infinitely many $k$ such that $T(n^k)>T(n^{k+1})$?
The original problem was solved (in the affirmative) by Beker [Be23b].
They also ask how many consecutive integers $>n$ can be represented as such a sum? Is it true that, for any $c>0$ at least $cn$ such integers are possible (for sufficiently large $n$)?
Note that $8$ is 3-full and $9$ is 2-full. Is the the only pair of such consecutive integers?
Unfortunately the problem is trivially true as written (simply taking $\{1,\ldots,k\}$ and $n>k^{1/\epsilon}$). There are (at least) two possible variants which are non-trivial, and it is not clear which Erdős and Graham meant. Let $P$ be the sequence of $k$ consecutive integers sought for. The potential strengthenings which make this non-trivial are:
See also [382].
See also [380].
Overholt [Ov93] has shown that this has only finitely many solutions assuming a weak form of the abc conjecture.
See also [401].
Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87].
Proved for all sets by Balasubramanian and Soundararajan [BaSo96].
See also [404].
Is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that if $p^{m_k}$ is the highest power of $p$ dividing $\sum_{i\leq k}a_i!$ then $m_k\to \infty$?
Erdős, Granville, Pomerance, and Spiro [EGPS90] have proved that the answer to the first two questions is yes, conditional on a form of the Elliott-Halberstam conjecture.
It is likely true that, if $k\to \infty$ however slowly with $n$, then for almost $n$ the largest prime factor of $\phi_k(n)$ is $\leq n^{o(1)}$.
Erdős and Graham report it could be attacked by sieve methods, but 'at present these methods are not strong enough'. Can one show that there exists an $\epsilon>0$ such that there are infinitely many $n$ where $m+\epsilon \nu(m)\leq n$ for all $m<n$?
The behaviour of $V(x)$ is now almost completely understood. Maier and Pomerance [MaPo88] proved \[V(x)=\frac{x}{\log x}e^{(C+o(1))(\log\log\log x)^2},\] for some explicit constant $C>0$. Ford [Fo98] improved this to \[V(x)\asymp\frac{x}{\log x}e^{C_1(\log\log\log x-\log\log\log\log x)^2+C_2\log\log\log x-C_3\log\log\log\log x}\] for some explicit constants $C_1,C_2,C_3>0$. Unfortunately this falls just short of an asymptotic formula for $V(x)$ and determining whether $V(2x)/V(x)\to 2$.
See also [417].
Erdős [Er73b] has shown that a positive density set of integers cannot be written as $\sigma(n)-n$.
This is true, as shown by Browkin and Schinzel [BrSc95], who show that any integer of the shape $2^{k}\cdot 509203$ is not of this form. It seems to be open whether there is a positive density set of integers not of this form.
See also [490].
Note that there are $\sim 2^{\binom{n}{2}}/n!$ many non-isomorphic graphs on $n$ vertices (folklore, often attributed to Pólya), and hence the bound in the problem statement is trivially best possible.
Erdős believed Brouwer's construction was essentially best possible, but Spencer suggested that $\gg \frac{2^{\binom{n}{2}}}{n!}$ may be possible. Erdős offered \$100 for a construction and \$25 for a proof that no such construction is possible.
Indeed, we apply this with $k=q=d$ and $a=1$ and let $p_m,\ldots,p_{m+{d-1}}$ be consecutive primes all congruent to $1$ modulo $d$, with $m>n+1$. If $p_{n+1}+\cdots+p_{m-1}\equiv r\pmod{d}$ with $1\leq r\leq d$ then \[d \mid p_{n+1}+\cdots +p_m+\cdots+p_{m+r-1}.\]
Elsholtz [El01] has proved there are no infinite sets $A,B,C$ such that $A+B+C$ agrees with the set of prime numbers up to finitely many exceptions.
See also [432].
The problem is written as Erdős and Graham describe it, but presumably they had in mind the regime where $n$ is fixed and $t\to \infty$.
Lagarias, Odlyzko, and Shearer [LOS83] proved this is sharp for the modular version of the problem; that is, if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ is such that $A+A$ contains no squares then $\lvert A\rvert\leq \tfrac{11}{32}N$. They also prove the general upper bound of $\lvert A\rvert\leq 0.475N$ for the integer problem.
In fact $\frac{11}{32}$ is sharp in general, as shown by Khalfalah, Lodha, and Szemerédi [KLS02], who proved that the maximal such $A$ satisfies $\lvert A\rvert\leq (\tfrac{11}{32}+o(1))N$.
See also [587].
This is true, as proved by Khalfalah and Szemerédi [KhSz06], who in fact prove the general result with $x+y=z^2$ replaced by $x+y=f(z)$ for any non-constant $f(z)\in \mathbb{Z}[z]$ such that $2\mid f(z)$ for some $z\in \mathbb{Z}$.
If $\delta'(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta'(n)=o(\delta(n))$?
Among many other results in [Fo08], Ford also proves that the second conjecture is false, and more generally that if $\delta_r(n)$ is the density of integers with exactly $r$ divisors in $(n,2n)$ then $\delta_r(n)\gg_r\delta(n)$.
See also [448].
Solved by Kleitman [Kl71], who proved \[\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.\]
A more precise result was proved by Hall and Tenenbaum [HaTe88] (see Section 4.6), who showed that the upper density is $\ll\epsilon \log(2/\epsilon)$. Hall and Tenenbaum further prove that $\tau^+(n)/\tau(n)$ has a distribution function.
Erdős and Graham also asked whether there is a good inequality known for $\sum_{n\leq x}\tau^+(n)$. This was provided by Ford [Fo08] who proved \[\sum_{n\leq x}\tau^+(n)\asymp x\frac{(\log x)^{1-\alpha}}{(\log\log x)^{3/2}}\] where \[\alpha=1-\frac{1+\log\log 2}{\log 2}=0.08607\cdots.\]
See also [448].
Is it true that, for any $0<\delta<1/2$, we have \[N(X,\delta)=o(X)?\] In fact, is it true that (for any fixed $\delta>0$) \[N(X,\delta)<X^{1/2+o(1)}?\]
Even a stronger statement is true, as shown by Konyagin [Ko01], who proved that \[N(X,\delta) \ll_\delta N^{1/2}.\]
Is there some $\delta>0$ such that \[\lim_{x\to \infty}N(X,\delta)=\infty?\]
What is the size of $D_n\backslash \cup_{m<n}D_m$?
If $f(N)$ is the minimal $n$ such that $N\in D_n$ then is it true that $f(N)=o(N)$? Perhaps just for almost all $N$?
Melfi [Me15] has proved that there are infinitely many primitive weird numbers, conditional on various well-known conjectures on the distribution of prime gaps. For example, it would suffice to show that $p_{n+1}-p_n <\frac{1}{10}p_n^{1/2}$ for sufficiently large $n$.
Watts has suggested that perhaps the obvious greedy algorithm defines such a permutation - that is, let $a_1=1$ and let \[a_{n+1}=\min \{ x : a_n+x\textrm{ is prime and }x\neq a_i\textrm{ for }i\leq n\}.\] In other words, do all positive integers occur as some such $a_n$? Do all primes occur as a sum?
As an indication of the difficulty, when $k=3$ the smallest $n$ such that $2^n\equiv 3\pmod{n}$ is $n=4700063497$.
The original formulation of this problem had an extra condition on the minimal element of the sequence $A_k$ being large, but Ryan Alweiss has pointed out that is trivially always satisfied since the minimal element of the sequence must grow by at least $1$ at each stage.
Find similar results for $\theta=\sqrt{m}$, and other algebraic numbers.
This was solved by Schinzel [Sc87], who proved that \[f(k) > \frac{\log\log k}{\log 2}.\] In fact Schinzel proves lower bounds for the corresponding problem with $P(x)^n$ for any integer $n\geq 1$, where the coefficients of the polynomial can be from any field with zero or sufficiently large positive characteristic.
Schinzel and Zannier [ScZa09] have improved this to \[f(k) \gg \log k.\]
See also [208].
See also [425].
This is true, and was proved by Szemerédi [Sz76].
This is true, and was proved by Margulis [Ma89].
Resolved by Kleitman [Kl69], who proved that the number of such families is \[2^{(1+o(1))\binom{n}{\lfloor n/2\rfloor}}.\]
This conjecture is true, and was proved by Marcus and Minc [MaMi62].
Erdős also conjectured the even weaker fact that there exists some $\sigma\in S_n$ such that $a_{i\sigma(i)}\neq 0$ for all $i$ and \[\sum_{i}a_{i\sigma(i)}\geq 1.\] This weaker statement was proved by Marcus and Ree [MaRe59].
van der Waerden's conjecture itself was proved by Gyires [Gy80], Egorychev [Eg81], and Falikman [Fa81].
If the sets $A_x$ are closed and have measure $<1$, then must there exist an independent set of size $3$?
Erdős writes in [Er61] that Gladysz has proved the existence of an independent set of size $2$ in the second question, but I cannot find a reference.
Hechler [He72] has shown the answer to the first question is no, assuming the continuum hypothesis.
Bannai, Bannai, and Stanton [BBS83] have proved that \[\lvert A\rvert \leq \binom{n+2}{2}.\] A simple proof of this upper bound was given by Petrov and Pohoata [PePo21].
Shengtong Zhang has observed that a simple lower bound of $\binom{n}{2}$ is given by considering all points with exactly two coordinates equal to $1$ and all others equal to $0$.
Sendov [Se93] provided the definitive answer, proving that $\alpha_N=\pi(1-1/n)$ for $2^{n-1}+2^{n-3}<N\leq 2^n$ and $\alpha_N=\pi(1-\frac{1}{2n-1})$ for $2^{n-1}<N\leq 2^{n-1}+2^{n-3}$.
The answer is in fact no in general, as shown by Kahn and Kalai [KaKa93], who proved that it is false for $n>2014$. The current smallest $n$ where Borsuk's conjecture is known to be false is $n=64$, a result of Brouwer and Jenrich [BrJe14].
If $\alpha(n)$ is the smallest number of pieces of diameter $<1$ required (so Borsuk's original conjecture was that $\alpha(n)=n+1$) then Kahn and Kalai's construction shows that $\alpha(n)\geq (1.2)^{\sqrt{n}}$. The best upper bound, due to Schramm [Sc88], is that \[\alpha(n) \leq (\sqrt{3/2}+o(1))^{n}.\] The true order of magnitude may be $\alpha(n)=c^n$ for some $c>1$.
Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$)?
Wintner [Wi44] proved that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2+o(1)},\] and Erdős improved the right-hand side to $N^{1/2}(\log N)^{O(1)}$. Lau, Tenenbaum, and Wu [LTW13] have shown that, almost surely, \[\sum_{m\leq N}f(m)\ll N^{1/2}(\log\log N)^{2+o(1)}.\] Harper [Ha13] has shown that the sum is almost surely not $O(N^{1/2}/(\log\log N)^{5/2+o(1)})$, and conjectured that in fact Erdős' conjecture is false, and almost surely \[\sum_{m\leq N}f(m) \ll N^{1/2}(\log\log N)^{1/4+o(1)}.\] This was proved by Caich [Ca23].
For any $t\in (0,1)$ let $t=\sum_{k=1}^\infty \epsilon_k(t)2^{-k}$ (where $\epsilon_k(t)\in \{0,1\}$). If $S_n(t)$ is the number of roots of $\sum_{1\leq k\leq n}\epsilon_k(t)z^k$ in $\lvert z\rvert \leq1$ then is it true that, for almost all $t\in (0,1)$, \[\lim_{n\to \infty}\frac{S_n(t)}{n}=\frac{1}{2}?\]
See also [521].
Kahane [Ka59] showed that $a_n=\frac{1+c}{n}$ with $c>0$ has this property, which Erd\H{s} (unpublished) improved to $a_n=\frac{1}{n}$. Erd\{o}s also showed that $a_n=\frac{1-c}{n}$ with $c>0$ does not have this property.
Solved by Shepp [Sh72], who showed that a necessary and sufficient condition is that \[\sum_n \frac{e^{a_1+\cdots+a_n}}{n^2}=\infty.\]
It is 'well known' that, for almost all $\epsilon_n=\pm 1$, the series diverges for almost all $\lvert z\rvert=1$ (assuming only $\sum \lvert a_n\rvert^2=\infty$).
Dvoretzky and Erdős [DE59] showed that if $\lvert a_n\rvert >c/\sqrt{n}$ then, for almost all $\epsilon_n=\pm 1$, the series diverges for all $\lvert z\rvert=1$.
Kesten [Ke63] proved that $C_k=2k-1-1/2k+O(1/k^2)$, and more precise asymptotics are given by Clisby, Liang, and Slade [CLS07].
Conway and Guttmann [CG93] showed that $C_2\geq 2.62$ and Alm [Al93] showed that $C_2\leq 2.696$.
Erdős and Spencer [ErSp89] proved that \[F(k) \geq 2^{ck^2/\log k}\] for some constant $c>0$. Balogh, Eberhrad, Narayanan, Treglown, and Wagner [BENTW17] have improved this to \[F(k) \geq 2^{2^{k-1}/k}.\]
Erdős writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that if $\omega(n)=k$ for all $n\in A$ then there must be some $a_1,\ldots,a_r\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erdős and Rado gives the upper bound $c_r^kk!$.
Erdős describes a construction of Ruzsa which disproves this: consider the set of all squarefree numbers of the shape $p_1\cdots p_r$ where $p_{i+1}>2p_i$ for $1\leq i<r$. This set has positive density, and hence if $A$ is its intersection with $(N/2,N)$ then $\lvert A\rvert \gg N$ for all large $N$. Suppose now that $p_1a_1=p_2a_2=p_3a_3$ where $a_i\in A$ and $p_1,p_2,p_3$ are distinct primes. Without loss of generality we may assume that $a_2>a_3$ and hence $p_2<p_3$, and so since $p_2p_3\mid a_1\in A$ we must have $2<p_3/p_2$. On the other hand $p_3/p_2=a_2/a_3\in (1,2)$, a contradiction.
See also [165] and the entry in the graphs problem collection.
Implied by [548].
Brandt and Dobson [BrDo96] have proved this for graphs of girth at least $5$. Wang, Li, and Liu [WLL00] have proved this for graphs whose complements have girth at least $5$. Saclé and Woznik [SaWo97] have proved this for graphs which contain no cycles of length $4$. Yi and Li [YiLi04] have proved this for graphs whose complements contain no cycles of length $4$.
This is false: Norin, Sun, and Zhao [NSZ16] have proved that if $T$ is the union of two stars on $k$ and $2k$ vertices, with an edge joining the centre of the two stars, then $R(T)\geq (4.2-o(1))k$. The best upper bound for the Ramsey number for this tree is $R(T)\leq 4.27492k+1$, obtained by Dubó and Stein [DuSt24].
This identity was proved for $k>n^2-2$ by Bondy and Erdős [BoEr73]. Nikiforov [Ni05] extended this to $k\geq 4n+2$.
Keevash, Long, and Skokan [KLS21] have proved this identity when $k\geq C\frac{\log n}{\log\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$.
Luczak [Lu99] has shown that $R(C_n;3)\leq (4+o(1))n$ for all $n$, and in fact $R(C_n;3)\leq 3n+o(n)$ for even $n$.
Kohayakawa, Simonovits, and Skokan [KSS05] proved this conjecture when $n$ is sufficiently large and odd. Benevides and Skokan [BeSk09] proved that if $n$ is sufficiently large and even then $R(C_n;3)=2n$.
If $G$ has $n$ vertices and maximum degree $d$ then prove that \[\hat{R}(G)\ll_d n.\]
This was disproved for $d=3$ by Rödl and Szemerédi [RoSz00], who constructed a graph on $n$ vertices with maximum degree $3$ such that \[\hat{R}(G)\gg n(\log n)^{c}\] for some absolute constant $c>0$. Tikhomirov [Ti22b] has improved this to \[\hat{R}(G)\gg n\exp(c\sqrt{\log n}).\] It is an interesting question how large $\hat{R}(G)$ can be if $G$ has maximum degree $3$. Kohayakawa, Rödl, Schacht, and Szemerédi [KRSS11] proved an upper bound of $\leq n^{5/3+o(1)}$ and Conlon, Nenadov, and Trujić [CNT22] proved $\ll n^{8/5}$. The best known upper bound of $\leq n^{3/2+o(1)}$ is due to Draganić and Petrova [DrPe22].
Determine \[\hat{R}(K_{n,n}),\] where $K_{n,n}$ is the complete bipartite graph with $n$ vertices in each component.
Conlon, Fox, and Wigderson [CFW23] have proved that, for any $s\leq t$, \[\hat{R}(K_{s,t})\gg s^{2-\frac{s}{t}}t2^s,\] and prove that when $t\gg s\log s$ we have $\hat{R}(K_{s,t})\asymp s^2t2^s$. They conjecture that this should hold for all $s\leq t$, and so in particular we should have $\hat{R}(K_{n,n})\asymp n^32^n$.
Let $F_1$ and $F_2$ be the union of stars. More precisely, let $F_1=\cup_{i\leq s} K_{1,n_i}$ and $F_2=\cup_{j\leq t} K_{1,m_j}$. Prove that \[\hat{R}(F_1,F_2) = \sum_{2\leq k\leq s+2}\max\{n_i+m_j-1 : i+j=k\}.\]
Prove that, for $r\geq 3$, \[\log_{r-1} R_r(n) \asymp_r n,\] where $\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm. That is, does $R_r(n)$ grow like \[2^{2^{\cdots n}}\] where the tower of exponentials has height $r-1$?
Prove that, for every $0\leq \alpha\leq 1/2$, \[F(n,\alpha)\sim c_\alpha\log n\] for some constant $c_\alpha$ depending only on $\alpha$.
Note that when $\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, and hence we recover the classical Ramsey numbers.
See also [161].
Is there some constant $c>0$ such that \[R_3(n) \geq 2^{2^{cn}}?\]
Is it true that \[R^*(G) \leq 2^{O(n)}\] for any graph $G$ on $n$ vertices?
Rödl [Ro73] proved this when $G$ is bipartite. Kohayakawa, Prömel, and Rödl [KPR98] have proved that \[R^*(G) < 2^{O(n(\log n)^2)}.\] An alternative (and more explicit) proof was given by Fox and Sudakov [FoSu08]. Conlon, Fox, and Sudakov [CFS12] have improved this to \[R^*(G) < 2^{O(n\log n)}.\]
The graph $H_5$ can also be described as $K_4^*$, obtained from $K_4$ by subdividing one edge. ($K_4$ itself is not Ramsey size linear, since $R(4,n)\gg n^{3-o(1)}$, see [166].) Bradać, Gishboliner, and Sudakov [BGS23] have shown that every subdivision of $K_4$ on at least $6$ vertices is Ramsey size linear, and also that $R(H_5,H) \ll m$ whenever $H$ is a bipartite graph with $m$ edges and no isolated vertices.
Erdős [Er64c] and Bondy and Simonovits [BoSi74] showed that \[\mathrm{ex}(n;C_{2k})\ll kn^{1+\frac{1}{k}}.\]
Benson [Be66] has proved this conjecture for $k=3$ and $k=5$. Lazebnik, Ustimenko, and Woldar [LUW95] have shown that, for arbitrary $k\geq 3$, \[\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{2}{3k-3+\nu}},\] where $\nu=0$ if $k$ is odd and $\nu=1$ if $k$ is even. See [LUW99] for further history and references.
See also [574] and the entry in the graphs problem collection.
See also [573] and the entry in the graphs problem collection.
Is it true that, for every $\mathcal{F}$, if there is a bipartite graph in $\mathcal{F}$ then there exists some bipartite $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]
See also [180] and the entry in the graphs problem collection.
Komlós and Sós conjectured the generalisation that if at least $n/2$ vertices have degree at least $k$ then $G$ contains any tree with $k$ vertices.
Frankl and Rödl [FrRo86] proved $N\leq 7\times 10^{11}$, which Spencer [Sp88] improved to $\leq 3\times 10^{9}$. This resolved the initial challenge of Erdős [Er75d] to beat $10^{10}$.
Lu [Lu07] proved $N\leq 9697$ vertices. The current record is due to Dudek and Rödl [DuRo08] who proved $N\leq 941$ vertices. For further information we refer to a paper of Radziszowski and Xu [RaXu07], who prove that $N\geq 19$ and speculate that $N\leq 127$.
Hajos [Lo68] has conjectured that if $G$ has all degrees even then $G$ can be partitioned into at most $\lfloor n/2\rfloor$ edge-disjoint cycles.
See also [184] and the entry in the graphs problem collection.
Fox and Sudakov [FoSu08b] have proved the second statement when $\delta >n^{-1/5}$.
Chakraborti, Janzer, Methuku, and Montgomery [CJMM24] have shown that such a graph can have at most $n(\log n)^{O(1)}$ many edges. Indeed, they prove that there exists a constant $C>0$ such that for any $k\geq 2$ there is a $c_k$ such that if a graph has $n$ vertices and at least $c_kn(\log n)^{C}$ many edges then it contains $k$ pairwise edge-disjoint cycles with the same vertex set.
Essentially solved by Nguyen and Vu [NgVu10], who proved that $\lvert A\rvert\ll N^{1/3}(\log N)^{O(1)}$.
See also [438].
This question was asked by Erdős to a young Terence Tao in 1985. We thank Tao for sharing this memory and a letter of Erdős describing the problem.
The restriction to $k\geq 4$ is necessary since Sylvester has shown that $f_3(n)= n^2/6+O(n)$. (See also Burr, Grünbaum, and Sloane [BGS74] and Füredi and Palásti [FuPa84] for constructions which show that $f_3(n)\geq(1/6+o(1))n^2$.)
For $k\geq 4$, Kárteszi [Ka] proved \[f_k(n)\gg_k n\log n.\] Grünbaum [Gr76] proved \[f_k(n) \gg_k n^{1+\frac{1}{k-2}}.\] Erdős speculated this may be the correct order of magnitude, but Solymosi and Stojaković [SoSt13] give a construction which shows \[f_k(n)\gg_k n^{2-O_k(1/\sqrt{\log n})}\]
Erdős thought that $g(n) \gg n$, but in fact $g(n)=o(n)$, which follows from the density Hales-Jewett theorem proved by Furstenberg and Katznelson [FuKa91] (see [185]).
This is true, and was proved by Chang [Ch72]. Milner modified his proof to prove that this remains true if we replace $K_3$ by $K_m$ for all finite $m<\omega$ (a shorter proof was found by Larson [La73]).
The first open case is $\beta=\omega^2$ (see [591]). Galvin and Larson [GaLa74] have shown that if $\beta\geq 3$ has this property then $\beta$ must be 'additively indecomposable', so that in particular $\beta=\omega^\gamma$ for some $\gamma<\omega_1$. Galvin and Larson conjecture that every $\beta\geq 3$ of this form has this property.
See also [590].
Every graph with chromatic number $\aleph_1$ contains all sufficiently large odd cycles (which have chromatic number $3$), see [594]. This was proved by Erdős, Hajnal, and Shelah [EHS74]. Erdős writes that 'probably' every graph with chromatic number $\aleph_1$ contains as subgraphs all graphs with chromatic number $4$ with sufficiently large girth.
See also [596].
Whether this is true for $G_1=K_4$ and $G_2=K_3$ is the content of [595].
This was proved by Aharoni and Berger [AhBe09].
Larson [La90] proved this is true for all $\alpha<2^{\aleph_0}$ assuming Martin's axiom.
It may be true that there are $\gg n$ many such points, or that this is true on average.
The best known bound is \[\gg n^{c-o(1)},\] due to Katz and Tardos [KaTa04], where \[c=\frac{48-14e}{55-16e}=0.864137\cdots.\]
Solved (for all sufficiently large $n$) completely by Erdős and Salamon [ErSa88]; the full description is too complicated to be given here.
Let $F(n)$ count the number of possible sets $A$ that can be constructed this way. Is it true that \[F(n) \leq \exp(O(\sqrt{n}))?\]
Erdős and Faudree observed that every graph with $2n$ vertices and at least $n^2+1$ edges has a triangle whose vertices are joined to at least $n+2$ vertices.
Day and Johnson [DaJo17] have shown that \[f(n)\geq 2^{c\sqrt{\log n}}\] for some constant $c>0$.
This would be best possible, since there exist triangle-free graphs with all independent sets of size $O(\sqrt{n\log n})$, which follows from the lower bound for $R(3,k)$ by Kim [Ki95] (see [165]).
Indeed, Erdős, Gallai, and Tuza speculate that if $f(n)$ is the largest $k$ such that every triangle-free graph on $n$ vertices contains an independent set on $f(n)$ vertices, then $\tau(G)\leq n-f(n)$.
In [Er94] Erdős asks for a weaker upper bound $\tau(G) \leq n-\omega(n)\sqrt{n}$ for any $\omega(n)\to \infty$.
See also [611], this entry and and this entry in the graphs problem collection.
Determine the best possible $t$ such that, if $G$ is an $r$-uniform hypergraph $G$ where every subgraph $G'$ on at most $3r-3$ vertices has $\tau(G')\leq 1$, we have $\tau(G)\leq t$.