8 problems

Let $R_3(n;3,2)$ be the least integer $N$ such that every three-coloring of the edges of the complete three-uniform hypergraph $K_N^{(3)}$ contains a copy of $K_n^{(3)}$ using at m…

Let $G$ be an $n$-vertex $r$-graph, let $s(G;m)$ denote the number of values of $f$ for which $G$ contains $m$ vertices spanning exactly $f$ edges, and let $g_r(m)$ be the threshol…

For a family $\mathcal{F}$ of $3$-graphs, define … Let $\mathcal{F}_{\mathrm{poly}}$, $\mathcal{F}_{\mathrm{part\text{-}poly}}$, $\mathcal{F}_{\mathrm{polyfact}}$, and…

Let $k\geq 10$ be an integer, and let $p$ and $q$ be integers satisfying $p\geq k+2$ and $q\geq k+1$. Write $r_k(p,q)$ for the smallest integer $n$ such that every red-blue colorin…

The constructive lower-bound conjecture. The displayed inequality holds for all such $r$, $t$, and hypergraphs $H_1,\dots,H_t$. The preceding constructive theorems provide evidence…

Let $r\ge 2$, let $T$ be an $r$-uniform tree, and let $n$ be a positive integer. An $r$-uniform tree $T$ is $n$-good when its Ramsey number against the complete $r$-uniform hypergr…

Let $r\ge 2$, let $T_1$ and $T_2$ be $r$-uniform trees of order $m$, and let $K_n^{(r)}$ be the complete $r$-uniform hypergraph on $n$ vertices. The Ramsey equivalence conjecture f…

Multiple-copy path and cycle conjecture. For every $k\geq 3$, $m\geq n$ and $r\geq s\geq 3$,