5 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 $\epsilon>0$. 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…