16 problems

Let $K_{\ell+1}^{(r)+}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices, let $M_{s+1}^{(r)+}$ be an $r$-uniform matching with $s+1$ edges, and let…

Let ${\mathrm{ex}}^{\sum}_r(n,k,K_\ell^{(r)+})$ denote the rainbow hyper-Turán number for the $r$-uniform expansion $K_\ell^{(r)+}$, and let $t_r(n,\ell-1)$ be the number of edges…

Let $K^{(r)+}_{\ell+1}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices, let $M^{(r)+}_{s+1}$ be an $r$-uniform matching with $s+1$ edges, and let…

Let $G$ be a graph with chromatic number $\chi(G)=\ell>r$, and let $U$ be an independent set of $G$ such that deleting $U$ results in a graph $G_0$ of chromatic number $\ell-1$. Su…

Let $\mathcal{T}_r(n,\ell)$ be the complete balanced $\ell$-partite $r$-graph, and let $K^{(r)+}_{\ell+1}$ be the $r$-uniform expansion of the complete graph on $\ell+1$ vertices.…

Let $K^{(r)+}_3$ denote the $r$-uniform expansion of the triangle, and let ${\mathrm{ex}}_r(n,K^{(r)+}_3)$ be its $r$-uniform Turán number. Erdős's expanded-triangle conjecture. Fo…

The codegree-squared density conjecture for $K_5^4$. The lower bound is tight:

The lower-bound sharpness conjecture for $K_5^<$. All four lower bounds are sharp:

Let $G$ be the iterated blow-up of the complement of the Fano plane. It is $J_4$-free and has codegree squared density at least $0.28$. The complement-of-Fano blow-up conjecture fo…

Let $G$ be the iterated blow-up of an edge, obtained from a complete balanced $3$-partite $3$-graph by inserting a complete balanced $3$-partite $3$-graph recursively in each part.…

Let $S_6$ be the $3$-graph on six vertices with edge set … Let $n=6^k$ and let $G$ be the iterated blow-up of $S_6$. The $3$-graph $G$ is $K_4^{3-}$-free and satisfies…

Let $\mathbb{F}$ be the Fano plane, and let $B_n$ be the complete balanced bipartite $3$-graph on $n$ vertices. For a $3$-graph $H$, write $\operatorname{co}_2(H)$ for its codegree…

For fixed integers $s$ and $t$, let $\mathcal D_r(s,t)$ consist of all $r$-sets $A$ containing a fixed $(r-t)$-set $P$ and contained in $P\cup Q$, where $Q$ is a fixed disjoint $s$…

Let $\mathcal A$ be a family of $3$-sets of $[n]$, where $n=7^k$, and suppose that $\mathcal A$ contains no daisy. The power-of-seven daisy bound. … This is proposed as the exact u…

Let $\mathcal D_3$ be the daisy formed from a fixed $1$-set and a disjoint $4$-set, and let $\pi(\mathcal D_3)$ be its limiting Turán density. The 3-daisy density conjecture. … The…

Let $[n]=\{1,\ldots,n\}$. For a set $X$, write $X^{(r)}$ for its $r$-subsets. An $r$-graph is a subset of $X^{(r)}$. Given an $(r-2)$-set $P$ and a disjoint $4$-set $Q$, the $r$-da…