If $G$ is not bipartite the answer is yes, proved by Erdős, Frankl, and Rödl [ErFrRo86]. The answer is no for $G=C_6$, the cycle on 6 vertices. Morris and Saxton [MoSa16] have proved there are at least \[2^{(1+c)\mathrm{ex}(n;C_6)}\] such graphs for infinitely many $n$, for some constant $c>0$. It is still possible (and conjectured by Morris and Saxton) that the weaker bound of \[2^{O(\mathrm{ex}(n;G))}\] holds for all $G$.

Conjectured by Erdős and Simonovits [ErSi84]. Erdős first offered \$250 for a proof and \$100 for a counterexample, but in [Er93] offered \$500 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 [ErSi84]. Open even for $r=2$. 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.

Conjectured by Erdős and Simonovits [ErSi84]. 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 [113], [146], and [714].

See also the entry in the graphs problem collection.

A problem of Erdős and Simonovits.

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 problem of Turán. Turán observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that \[\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.\] This is probably the truth. The current best upper bound is \[\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},\] due to Razborov [Ra10].

See also [712] for the general case.

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 extremal 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.

It is easy to see that $\mathrm{ex}(n;C_{2k+1})=\lfloor n^2/4\rfloor$ for any $k\geq 1$ (and $n>2k+1$) (since no bipartite graph contains an odd cycle). Erdős and Klein [Er38] proved $\mathrm{ex}(n;C_4)\asymp n^{3/2}$.

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 [765] and the entry in the graphs problem collection.

A problem of Erdős and Simonovits, who proved that \[\mathrm{ex}(n;\{C_4,C_5\})=(n/2)^{3/2}+O(n).\]

See also [574] and the entry in the graphs problem collection.

A problem of Erdős and Simonovits.

See also [573] and the entry in the graphs problem collection.

A problem of Erdős and Simonovits.

See also [180] and the entry in the graphs problem collection.

Erdős and Simonovits [ErSi70] proved that \[(\tfrac{1}{2}+o(1))n^{3/2}\leq \mathrm{ex}(n;Q_3) \ll n^{8/5}.\] In [Er81] and [Er93] Erdős asked whether it is $\asymp n^{8/5}$.

A theorem of Sudakov and Tomon [SuTo22] implies \[\mathrm{ex}(n;Q_k)=o(n^{2-\frac{1}{k}}).\] Janzer and Sudakov [JaSu22b] have improved this to \[\mathrm{ex}(n;Q_k)\ll_k n^{2-\frac{1}{k-1}+\frac{1}{(k-1)2^{k-1}}}.\] See also 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.

Turán proved that, when $r=2$, this limit is \[\frac{1}{2}\left(1-\frac{1}{k-1}\right).\] Erdős [Er81] offered \$500 for the determination of this value for any fixed $k>r>2$, and \$1000 for 'clearing up the whole set of problems'.

See also [500] for the case $r=3$ and $k=4$.

A problem of Erdős and Simonovits. This is not true for hypergraphs, as shown by Ruzsa and Szemerédi [RuSz78], who proved that if $G$ is a $3$-uniform hypergraph on $6$ vertices containing $3$ $3$-edges then $\mathrm{ex}_3(n;G)=o(n^2)$ and yet for every $\epsilon >0$ \[\mathrm{ex}_3(n;G) >n^{2-\epsilon}\] for all large enough $n$.

See also [571].

Kövári, Sós, and Turán [KST54] proved \[\mathrm{ex}(n; K_{r,r}) \ll n^{2-1/r}.\] Brown [Br66] proved the conjectured lower bound when $r=3$.

See also [147].

Erdős and Klein [Er38] proved $\mathrm{ex}(n;C_4)\asymp n^{3/2}$. Reiman [Re58] proved \[\frac{1}{2\sqrt{2}}\leq \lim \frac{\mathrm{ex}(n;C_4)}{n^{3/2}}\leq \frac{1}{2}.\] Erdős and Rényi, and independently Brown, gave a construction that proved if $n=q^2+q+1$, where $q$ is a prime power, then \[\mathrm{ex}(n;C_4)\geq\frac{1}{2}q(q+1)^2.\] Coupled with the upper bound of Reiman this implies that $\mathrm{ex}(n;C_4)\sim\frac{1}{2}n^{3/2}$ for all large $n$. Füredi [Fu83] proved that if $q>13$ then \[\mathrm{ex}(n;C_4)=\frac{1}{2}q(q+1)^2.\]

See also [572].

Dirac and Erdős proved independently that when $l=\lfloor k^2/4\rfloor+1$ \[f(n;k,l)\leq \lfloor n^2/4\rfloor+1.\]

Czipszer proved that $g_k(n)$ exists for all $k$, and in fact $g_k(n)\leq (k+1)n$. Erdős wrote it is 'easy to see' that \[g_k(n)\geq (k+1)n-(k+1)^2.\] Pósa proved that $g_1(n)=2n-4$ for $n\geq 4$. Erdős could prove the conjectured equality for $n\geq 2k+2$ when $k=2$ or $k=3$.

The conjectured equality was proved for $n\geq 3k+3$ by Jiang [Ji04].

Curiously, in [Er69b] Erdős mentions this problem, but states that his conjectured equality for $g_k(n)$ was disproved (for general $k$) by Lewin, citing oral communication. Perhaps Lewin only disproved this for small $n$, or perhaps Lewin's disproof was simply incorrect.

Balogh has observed that this problem is probably misstated by Erdős - indeed, every graph with $5$ vertices spanning $7$ edges contains a graph on $4$ vertices spanning $3$ edges, so the second condition can be dropped.

This problem is then now asking how many edges a $3$-uniform hypergraph can have before it contains $K_4$ minus an edge, and whether the critical edge density is $2/9$. In fact there is a construction of Frankl and Füredi [FrFu84] showing it must be at least $2/7$, which is the conjectured truth (although Turán conjectured before [Er69] that that the edge density was $1/4$, and so likely there is simply a typo in this problem's statement).

Harris has provided the following simple counterexample to the problem as stated: the $3$-uniform graph on $\{1,\ldots,9\}$ with $28$ edges, formed by taking $27$ edges by choosing one element each from $\{1,2,3\},\{4,5,6\},\{7,8,9\}$, and then adding the edge $\{1,2,3\}$.