11 problems

Let $Q_7$ be the 7-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_7,C_4)$ for the maximum number of edges in a subgraph of $Q_7$ containi…

Let $Q_8$ be the 8-dimensional hypercube and let $C_4$ denote the cycle of length four. Write $\mathrm{ex}(Q_8,C_4)$ for the maximum number of edges in a subgraph of $Q_8$ containi…

Verstraëte's conjecture. There exists a positive constant $c$ such that every $C_{2k}$-free bipartite graph $G$ has a $C_{2\ell}$-free subgraph $F$ satisfying

Let $h$ be even, let $p_0$ be the parameter from the paper, and let $p\in(0,p_0]$. Write $\operatorname{ed}_{\operatorname{Forb}(C_h)}(p)$ for the edit-distance function for forbid…

Let $k\geq3$, and let $G$ be a $C_k$-free graph with $m$ edges. Write $\operatorname{Max-Cut}(G)$ for the maximum number of edges in a bipartite subgraph of $G$. Alon's conjecture.…

Let $n$ be the number of vertices and let $C_{2k}$ be the cycle of length $2k$. Write $\operatorname{ex}^*(n,C_{2k})=\operatorname{ex}(n,P_1,\textnormal{rainbow-}C_{2k})$ for the m…

Let $Q_n$ be the $n$-dimensional hypercube, and let $G$ be a quadrilateral-free subgraph of $Q_n$. Erdős's conjecture. As $n$ tends to infinity, $G$ has asymptotically at most half…

Let $C_4$ denote the cycle on four vertices, and let $\mathrm{Ex}(n,C_4)$ be the family of extremal $C_4$-free graphs on $n$ vertices. An induced subgraph is obtained by selecting…

Let $\epsilon>0$. A countable infinite graph with no cycle of length four is called $C_4$-free, and let $\mathcal{G}_n$ denote the subgraph induced by the first $n$ vertices. Write…

Let $\mathcal{G}$ be a countable infinite graph with no cycle of length four, and let $\mathcal{G}_n$ be the subgraph induced by its first $n$ vertices. Write…

Let $C_k$ denote a cycle of length $k$. Let $\operatorname{ex}(n,\{C_3,C_4\})$ be the maximum number of edges in a graph on $n$ vertices containing no triangle or four-cycle, and l…