Erdős called this 'perhaps my first serious problem'. The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdős and Moser [Er56] proved \[ N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.\] (In [Er85c] Erdős offered \$100 for any improvement of the constant $1/4$ here.)

A number of improvements of the constant have been given (see [St23] for a history), with the current record $\sqrt{2/\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound $N\geq \binom{n}{\lfloor n/2\rfloor}$.

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.)

This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink".

The sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.

See also [350].

A problem of Erdős and Turán. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Both proofs in fact give \[h(N) \leq N^{1/2}+N^{1/4}+1.\] Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to $0.998N^{1/4}$, which has been further optimised by O'Bryant [OB22] to yield \[h(N)\leq N^{1/2}+0.99703N^{1/4}\] for sufficiently large $N$.

See also [241] and [840].

The trivial greedy construction achieves $\gg N^{1/3}$. The current best bound of $\gg N^{\sqrt{2}-1+o(1)}$ is due to Ruzsa [Ru98]. (Erdős [Er73] had offered \$25 for any construction which achieves $N^{c}$ for some $c>1/3$.) Erdős proved that for every infinite Sidon set $A$ we have \[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0,\] and also that there is a set $A\subset \mathbb{N}$ with $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ such that $1_A\ast 1_A(n)=O(1)$.

Erdős and Rényi have constructed, for any $\epsilon>0$, a set $A$ such that \[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\] for all large $N$ and $1_A\ast 1_A(n)\ll_\epsilon 1$ for all $n$.

Erdős proved that if the pairwise sums $a+b$ are all distinct aside from the trivial coincidences then \[\liminf \frac{\lvert A\cap \{1,\ldots,N\}\rvert}{N^{1/2}}=0.\]

See also [707].

The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of \[\lvert A\rvert^2 \exp\left(-c\frac{\log\lvert A\rvert}{\log\log \lvert A\rvert}\right)\] for some constant $c>0$. The lower bound has been improved a number of times. The current record is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}\] due to Rudnev and Stevens [RuSt22] (note $1558/1167=1.33504\cdots$).

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, in [ErSz83] (and in [Er91]) Erdős and Szemerédi also conjecture that for any $m\geq 2$ and finite set of integers $A$ \[\max( \lvert mA\rvert,\lvert A^m\rvert)\gg \lvert A\rvert^{m-o(1)}.\] See [53] for more on this generalisation and [808] for a stronger form of the original conjecture. See also [818] for a special case.

Asked by Erdős and Szemerédi [ErSz83]. Solved in this form by Chang [Ch03].

Erdős and Szemerédi proved that there exist arbitrarily large sets $A$ such that the integers which are the sum or product of distinct elements of $A$ is at most \[\exp\left(c (\log \lvert A\rvert)^2\log\log\lvert A\rvert\right)\] for some constant $c>0$.

See also [52].

The best known result is due to Balogh, Hu, Lidicky, and Liu [BHLL14], who proved that $0.6068 n2^{n-1}$ edges suffice. Erdős [Er91] observes that there exist subgraphs with \[\geq \left(\frac{1}{2}+\frac{c}{n}\right)n2^{n-1}\] many edges without a $C_4$ (for some constant $c>0$). He suggests that it 'perhaps not hopeless' to determine the threshold exactly.

A similar question can be asked for other even cycles.

See also [666] and the entry in the graphs problem collection.

Conjectured by Erdős and Simonovits. Erdős offered \$250 for a proof and \$100 for a counterexample. Disproved by Janzer [Ja21], who constructed, for any $\epsilon>0$, a $3$-regular bipartite graph $H$ such that \[\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}.\]

See also [146] and [147] and the entry in the graphs problem collection.

Conjectured by Erdős and Simonovits. Open even for $r=3$. Alon, Krivelevich, and Sudakov [AKS03] have proved \[\mathrm{ex}(n;H) \ll n^{2-1/4r}.\] They also prove the full Erdős-Simonovits conjectured bound if $H$ is bipartite and the maximum degree in one component is $r$.

See also [113] and [147].

See also the entry in the graphs problem collection.

Spencer [Sp77] proved \[R(4,k) \gg (k\log k)^{5/2}.\] Ajtai, Komlós, and Szemerédi [AKS80] proved \[R(4,k) \ll \frac{k^3}{(\log k)^2}.\] This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that \[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]

This problem is #5 in Ramsey Theory in the graphs problem collection.

A problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi, who could prove this is true for $\delta>1/16$, and could further prove it for $\delta>0$ if we replace $K_5$ with $K_4$.

They further observed that it fails for $\delta =1/4$ if we replace $K_5$ with $K_7$: by a construction of Erdős and Rogers [ErRo62] (see [620]) there exists some constant $c>0$ such that, for all large $n$, there is a graph on $n$ vertices which contains no $K_4$ and every set of at least $n^{1-c}$ vertices contains a triangle. If we take two vertex disjoint copies of this graph and add all edges between the two copies then this yields a graph on $2n$ vertices with $\geq n^2$ edges, which contains no $K_7$, yet every set of at least $2n^{1-c}$ vertices contains a triangle.

See also [579] and the entry in the graphs problem collection.

A problem of Erdős and Simonovits. Bukh and Conlon [BuCo18] proved the first problem holds if we weaken asking for the extremal number of a single graph to asking for the extreaml number of a finite family of graphs.

A rational $\alpha\in [1,2]$ for which the first problem holds is known as a Turán exponent. Known Turán exponents are:

• $\frac{3}{2}-\frac{1}{2s}$ for $s\geq 2$ (Conlon, Janzer, and Lee [CJL21]).
• $\frac{4}{3}-\frac{1}{3s}$ and $\frac{5}{4}-\frac{1}{4s}$ for $s\geq 2$ (Jiang and Qiu [JiQi20]).
• $2-\frac{a}{b}$ for $\lfloor b/a\rfloor^3 \leq a\leq \frac{b}{\lfloor b/a\rfloor+1}+1$ (Jiang, Jiang, and Ma [JJM20]).
• $2-\frac{a}{b}$ with $b>a\geq 1$ and $b\equiv \pm 1\pmod{a}$ (Kang, Kim, and Liu [KKL21]).
• $1+a/b$ with $b>a^2$ (Jiang and Qiu [JiQi23]),
• $2-\frac{2}{2b+1}$ for $b\geq 2$ or $7/5$ (Jiang, Ma, and Yepremyan [JMY22]).
• $2-a/b$ with $b\geq (a-1)^2$ (Conlon and Janzer [CoJa22]).

See also [713] and the entry in the graphs problem collection.

A problem of Erdős, Hajnal, Sós, and Szemerédi, who could prove this is true for $\delta>1/8$.

See also [533] and the entry in the graphs problem collection.

Faudree proved that this is true if $G$ has $2n+1$ vertices.

See also the entry in the graphs problem collection.

A problem of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS93]. In other words, if $\mathrm{rt}(n;k,\ell)$ is the Ramsey-Turán number then is it true that \[\mathrm{rt}(n; 4,n/\log n)< (1/8-c)n^2?\] Erdős, Hajnal, Sós, and Szemerédi [EHSS83] proved that for any fixed $\epsilon>0$ \[\mathrm{rt}(n; 4,\epsilon n)< (1/8+o(1))n^2.\] Sudakov [Su03] proved that \[\mathrm{rt}(n; 4,ne^{-f(n)})=o(n^2)\] whenever $f(n)/\sqrt{\log n}\to \infty$.

Resolved by Fox, Loh, and Zhao [FLZ15] who showed that the answer is no; in fact they prove that \[\mathrm{rt}(n; 4, ne^{-f(n)})\geq (1/8-o(1))n^2\] whenever $f(n) =o(\sqrt{\log n/\log\log n})$.

See also [22] and the entry in the graphs problem collection.

In [Er91] Erdős further suggests that perhaps, for every $k\geq 3$, there are constants $c$ and $a_k<1$ such that every subgraph with at least $cn^{a_k}2^n$ many edges contains a $C_{2k}$, where $a_k\to 0$ as $k\to \infty$.

The answer to this problem is no: Chung [Ch92] and Brouwer, Dejter, and Thomassen [BDT93] constructed an edge-partition of $Q_n$ into four subgraphs, each containing no $C_6$.

See also [86].

See also [44] and [329].

A question of Erdős and Hajnal. Alon and Sudakov [AlSu07] proved that in fact \[\frac{(\log n)^2}{\log\log n}\ll f((\log n)^2,n) \ll (\log n)^2\] and \[f((\log n)^3,n)\asymp \frac{(\log n)^2}{\log\log n}.\]

See also [805].

A problem of Erdős and Hajnal, who thought that there is no such graph for $g(n)=(\log n)^3$. Alon and Sudakov [AlSu07] proved that there is no such graph with \[g(n)=\frac{c}{\log\log n}(\log n)^3\] for some constant $c>0$.

Alon, Bucić, and Sudakov [ABS21] construct such a graph with \[g(n)\leq 2^{2^{(\log\log n)^{1/2+o(1)}}}.\] See also [804].

A problem of Erdős and Szemerédi, which strengthens the conjecture [52].

This strong conjecture was disproved by Alon, Ruzsa, and Solymosi [ARS20], who constructed (for arbitrarily large $n$) a set of integers $A$ with $\lvert A\rvert=n$ and a graph $G$ with $\gg n^{5/3-o(1)}$ many edges such that \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \ll \lvert A\rvert^{4/3+o(1)}.\] Alon, Ruzsa, and Solymosi do prove, however, that if $A$ has size $n$ and $G$ has $m$ edges then \[\max(\lvert A+_GA\rvert,\lvert A\cdot_G A\rvert) \gg m^{3/2}n^{-7/4}.\]

A problem of Burr, Erdős, Graham, and Sós, who proved that \[F_k(n)\gg n^2.\]

See also [810].

A problem of Burr, Erdős, Graham, and Sós.

See also [809].

A problem of Erdős, Pyber, and Tuza, who observe that if $d_G(n)$ exists then $d_G(n) < \frac{n-1}{e(G)}$.

The Kürschák competition in Hungary in 1986 asked students to prove that $d_{K_3}(n)$ exists. Kostochka proved that $d_{K_3}(n)=n/4$ is the best possible. Tuza proved that \[d_{C_4}(n) \leq \left(\frac{1}{4}-c\right)n\] for some constant $c>0$. Brightwell and Trotter proved that \[d_{C_6}(n) > (1-o(1))\frac{n}{6}.\]

Burr, Erdős, Faudree, and Schelp [BEFS89] proved that \[R(n+1)-R(n) \geq 4n-8\] for all $n\geq 2$. The lower bound of [165] implies that \[R(n+2)-R(n) \gg n^{2-o(1)}.\]

A problem of Erdős and Hajnal, who could prove that \[n^{1/3}\ll h(n) \ll n^{1/2}.\]

Bucić and Sudakov [BuSu23] have proved \[h(n) \gg n^{5/12-o(1)}.\]

The case $k=3$ was a problem of Erdős and Hajnal [Er91]. The question for general $k$ was a conjecture of Erdős, Faudree, Rousseau, and Schelp [EFRS90], who proved that such a subgraph exists with at most $n-c_k\sqrt{n}$ vertices. Mousset, Noever, and Skorić [MNS17] improved this to \[n-c_k\frac{n}{\log n}.\] The full conjecture was proved by Sauermann [Sa19], who proved this with $c_k \gg 1/k^3$.

A problem of Erdős and Hajnal, who could prove it for $3\leq k\leq 6$.

A problem of Erdős and Hajnal. The example of $K_{n,n+1}$ shows that this fails if we only have $n^2+n$ edges.

A problem of Erdős and Sárközy who proved \[g_3(n) \gg \frac{3^n}{n^{O(1)}}.\]

This was proved by Solymosi [So09d], in the strong form \[\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.\]

See also [52].

Erdős and Freud [ErFr91] proved \[\left(\frac{3}{8}-o(1)\right)N \leq f(N) \leq \left(\frac{1}{2}+o(1)\right)N,\] and note that it is closely connected to the size of the largest quasi-Sidon set (see [840]).