18 problems

Erdős–Hajnal–Rado conjecture. There is a constant $c>0$ such that

Polynomial-or-exponential growth conjecture. For any fixed $s,t,k$, the function $r_k(s,t;n)$ grows either polynomially or exponentially in $n$.

Iterated-tripartite characterization. The Ramsey number $r(F,n)$ is polynomial in $n$ if and only if $F$ is iterated tripartite.

The folklore linear-3-graph conjecture. Every linear $3$-uniform hypergraph $H$ satisfies

The iterated-partite characterization. $H$ is iterated $k$-partite if and only if

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 $\mathcal{F}_{\mathrm{polyfact}}$ be the family of $3$-graphs $H$ satisfying $r(H,K_n^{(3)})\leq n^{O(n)}$. Non-polynomial-factorial class conjecture. There exist $3$-graphs th…

Let $G$ be a non-bipartite graph, let $L_G$ be its link $3$-uniform hypergraph, and let $r(L_G,K_n^{(3)})$ denote the corresponding hypergraph Ramsey number. Asymptotic constant co…

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 $P_3^{(3)}$ be the 3-uniform loose path of length $3$, and let $R(P_3^{(3)},r)$ be the minimum $n$ such that every $r$-edge-coloring of the complete 3-uniform hypergraph…

Let $P_\ell^{(k)}$ be the $k$-uniform loose path of length $\ell$, and let $R(P_\ell^{(k)},2)$ denote the minimum $n$ such that every red-blue edge-coloring of $K_n^{(k)}$ contains…

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…

For fixed $r\geq 2$, let $T^{(r)}$ be the $r$-uniform triangle consisting of edges $b_1,\ldots,b_r,a$, where $b_i\cap b_j=R$ for all $i<j$, $|R|=r-1$, and $a=\bigcup_i(b_i-R)$. Let…

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