27 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…

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…

For a triple system $F$, let $m_k(F)$ be the minimum number of edges in an $F$-free triple system with chromatic number at least $k$. Bohman–Frieze–Mubayi conjecture. There exists…

Let $H$ be a 3-uniform hypergraph on $n$ vertices, let $d_1(H)$ denote its first skeletal degeneracy, and let $r(H;q)$ be its $q$-colour Ramsey number. Define the double exponentia…

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

Erdős–Hajnal tower-equivalence conjecture. The two functions have the same tower growth rate:

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 fixed $k\geq2$, let $T^{(k)}$ be the $k$-uniform triangle consisting of $k+1$ edges in which $k$ edges share a common $(k-1)$-set and the remaining edge contains the remaining…

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…

For integers $n\geq4$, let $r_3(n,n)$ denote the diagonal Ramsey number for 3-uniform hypergraphs. Erdős's conjecture. There is an absolute constant $c>0$ such that … This is the c…

Let $k\geq 4$ and define the tower function by $\operatorname{twr}_1(x)=x$ and $\operatorname{twr}_{i+1}(x)=2^{\operatorname{twr}_i(x)}$. The diagonal hypergraph Ramsey number…

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…

Let $r_3(n,n)$ be the diagonal Ramsey number for 3-uniform hypergraphs, namely the least $N$ such that every red-blue coloring of the triples of an $N$-element set contains a monoc…

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…

Strengthened Erdős–Hajnal conjecture. For fixed $k\geq 4$,

Kostochka–Mubayi–Verstraëte conjecture.