22 problems

Let $H$ be a $3$-uniform hypergraph, let $K_n^3$ be the complete $3$-uniform hypergraph on $n$ vertices, and let $r(H,K_n^3)$ be the corresponding Ramsey number. Call $H$ iterated…

Let $g_3(s)$ denote the maximum number of edges in an $s$-vertex iterated blowup of a $3$-uniform edge, and let $r_3(s,e;t)$ be the corresponding hypergraph Ramsey number. Conjectu…

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of an edge. Thus $g_k(n)=0$ for $n<k$, and for $n\geq k$, … where the maximum is over compositions of…

Let $K^{<(k)}_n$ denote the ordered complete $k$-uniform hypergraph, let $MP^{<(k)}_{k+1}$ denote the monotone $(k+1)$-vertex path in the $k$-uniform setting, and let $t_{k-1}$ be…

Tight tree Ramsey growth conjecture. For $r\geq 5$, if $T$ is a non-trivial tight $r$-tree, then there exist constants $c_1,c_2>0$ such that, for every positive integer $n$,

The family-level tower lower-bound conjecture. If $k>s$, then there exists a positive constant $c$ such that

The tower lower-bound conjecture. If $k>s$ and $H$ is an $s$-tightly connected $k$-graph that is not $k$-partite, then there exists a positive constant $c$ such that

The iterated-tripartite classification conjecture. For a 3-graph $H$, there exists a constant $c$ depending only on $H$ such that

A $3$-graph $F$ is linear if any two of its edges share at most one vertex. For fixed $F$, let $r(F,K_n^{(3)})$ be the least $N$ such that every red-blue coloring of $K_N^{(3)}$ co…

For a fixed $3$-graph $H$, write $r(H,K_n^{(3)})$ for the least $N$ such that every red-blue coloring of the edges of $K_N^{(3)}$ contains a red copy of $H$ or a blue copy of…

A pair construction of a $3$-graph coloring is a coloring of $K_N^{(3)}$ obtained from functions $f:\binom{[N]}{2}\to[p]$ and $g:[p]^3\to\{\operatorname{red},\operatorname{blue}\}$…

Size-version of the Erdős–Hajnal conjecture. There is a positive constant $\epsilon$ such that every colouring $c$ of $K_n$ in colours from $[s]$ avoiding $T$ satisfies

For integers $3\leq t\leq k$, let $\overline r_k(k+1,t;P_n)$ be the least $N$ such that every red/blue coloring of the $k$-sets of an $N$-vertex set contains a blue tight path…

For fixed $k\geq4$, let $P_{k+1}$ denote the relevant $k$-uniform tight path, and let $\overline r_k(P_{k+1},n)$ be its Ramsey number against a complete $k$-uniform hypergraph on…

For each $k\geq3$, let $LC_5^{(k)}$ be the $k$-uniform loose 5-cycle, and let $r_k(LC_5,n)$ denote its Ramsey number against a complete $k$-uniform hypergraph on $n$ vertices. Kost…

For fixed $k\geq3$, let $LC_3^{(k)}$ be the $k$-uniform loose 3-cycle, whose consecutive edges intersect in exactly one vertex and whose nonconsecutive edges are disjoint. Write…

For integers $k\geq3$ and $2\leq t\leq k$, let $r_k(k+1,t;n)$ be the Erdős–Hajnal Ramsey parameter. Define $\operatorname{twr}_1(x)=x$ and…

For integers $4\leq k<s$ and $n$, let $r_k(s,n)$ be the least $N$ such that every red/blue coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a r…

Let $r_4(5,n)$ be the off-diagonal Ramsey number for 4-uniform hypergraphs, the least $N$ such that every red-blue coloring of the 4-edges of an $N$-vertex complete hypergraph cont…

For integers $2\le k < s < n$ and $2\le t\le {s\choose k}$, let $r_k(s,t;P_n)$ be the minimum $N$ such that every red/blue coloring of the $k$-sets of $[N]$ contains a monochromati…

For a positive integer $s$, let $F_1(s)$ be the maximum number of triples $(a,b,c)$ with $1\leq a<b<c\leq s$ satisfying a prescribed three-coloring pattern on the edges $(a,b)$,…

Let $\mathcal{F}$ be the minimal family of 3-uniform hypergraphs generated from the empty hypergraphs on $1$ and $2$ vertices and an edge by the operation $H(G,v)$. Let…