9 problems

Let $\mathfrak{S}_n$ be the symmetric group on $[n]=\{1,\dots,n\}$. For a fixed permutation $\pi$, let $\operatorname{Av}_n(\pi)$ be the set of permutations in $\mathfrak{S}_n$ tha…

Let $P\in\{0,1\}^{k\times l}$ be an acyclic forbidden pattern, and let $G(P)$ be its unordered bipartite graph. Füredi–Hajnal acyclic-pattern conjecture. One should have … The pape…

Let $P\in\{0,1\}^{k\times l}$ be any forbidden pattern, and let $G(P)$ be its unordered bipartite graph. Let $\operatorname{Ex}(P,n)$ denote the maximum number of ones in an…

Pettie–Tardos conjecture. Every light $\mathcal Q_{\mathrm{PT}}$-free pattern has a linear extremal function.

Let $\mathcal{P}_{\text{acyclic}}$ be the class of acyclic 0–1 patterns. Acyclic-pattern conjecture. Both of the following forms are proposed: - Weak Form: For all…

Let $F$ be an $r$-uniform forest with interval chromatic number $r$. Füredi–Jiang–Kostochka–Mubayi–Verstraëte conjecture. The maximum number of edges in a vertex-ordered $r$-unifor…

Acyclic forbidden-matrix conjecture. Let $P$ be an acyclic zero-one matrix. Then every $n\times n$ zero-one matrix that is both $P$-free and $P^c$-free contains a $cn\times cn$ hom…

Acyclic-matrix conjecture. For every acyclic zero-one matrix $P$ and every $\varepsilon>0$, there is a $\delta>0$ such that every $P$-free $n\times n$ zero-one matrix with at least…

Simple-matrix conjecture. Let $P$ be a simple zero-one matrix. Then every $P$-free $n\times n$ zero-one matrix contains a $cn\times cn$ homogeneous submatrix, for a suitable consta…