3 solved out of 13 shown

Is it true that the number of graphs on $n$ vertices which do not contain $G$ is
\[\leq 2^{(1+o(1))\mathrm{ex}(n;G)}?\]

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 $G$.

$250

If $G$ is bipartite then $\mathrm{ex}(n;G)\ll n^{3/2}$ if and only $G$ is $2$-degenerate, that is, $G$ contains no induced subgraph with minimal degree at least 3.

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.

$500

If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then
\[\mathrm{ex}(n;H) \ll n^{2-1/r}.\]

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 $r$.

$500

If $H$ is bipartite and is not $r$-degenerate, that is, there exists an induced subgraph of $H$ with minimum degree $>r$ then
\[\mathrm{ex}(n;H) > n^{2-\frac{1}{r}+\epsilon}.\]

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

If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$.

Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]

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.

What is $\mathrm{ex}_3(n,K_4^3)$? That is, the largest number of $3$-edges which can placed on $n$ vertices so that there exists no $K_4^3$, a set of 4 vertices which is covered by all 4 possible $3$-edges.

A problem of Turan. Turan 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].

Show that for any rational $\alpha \in (1,2)$ there exists a bipartite graph $G$ such that
\[\mathrm{ex}(n;G)\asymp n^{\alpha}.\]
Conversely, if $G$ is bipartite then must there exist some rational $\alpha$ such that\[\mathrm{ex}(n;G)\asymp n^{\alpha}?\]

A problem of Erdős and Simonovits.

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). It is also known that $\mathrm{ex}(n;C_4)\asymp n^{3/2}$.

Erdős [Er65] 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.

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.

Is it true that, for $k\geq 2$,
\[\mathrm{ex}(n;\{C_{2k-1},C_{2k}\})=(1+o(1))(n/2)^{1+\frac{1}{k}}.\]

A problem of Erdős and Simonovits.

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

If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$.

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 \[\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?\]

A problem of Erdős and Simonovits.

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

Let $Q_k$ be the $k$-dimensional hypercube graph (so that $Q_k$ has $2^k$ vertices and $k2^{k-1}$ edges). Determine the behaviour of
\[\mathrm{ex}(n;Q_k).\]

Erdős and Simonovits [ErSi70] proved that
\[(\tfrac{1}{2}+o(1))n^{3/2}\leq \mathrm{ex}(n;Q_3) \ll n^{8/5}.\]
A theorem of Sudakov and Tomon [SuTo22] implies
\[\mathrm{ex}(n;Q_k)=o(n^{2-\frac{1}{k}}).\]
Janzer and Sudakov [JaSu24] 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.

Let $\epsilon>0$ and $n$ be sufficiently large. Show that, if $G$ is a graph on $n$ vertices which does not contain $K_{2,2,2}$ and $G$ has at least $\epsilon n^2$ many edges, then $G$ contains an independent set on $\gg_\epsilon n$ many vertices.

A problem of Erdős, Hajnal, Sós, and Szemerédi.