Hypergraph Ramsey numbers of cliques versus stars

Abstract

Let denote the complete 3‐uniform hypergraph on vertices and the 3‐uniform hypergraph on vertices consisting of all edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off‐diagonal Ramsey number exhibits an unusual intermediate growth rate, namely, for some positive constants and . The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum such that any 2‐edge‐coloring of the Cartesian product contains either a red rectangle or a blue ?

A lower bound for set‐coloring Ramsey numbers

“…such that F ⊂ B. 5 We will not be able to prove this using a simple first moment argument, summing over all graphs F ∈  , since the probability of the event {F ⊂ B} is not always sufficiently small. Instead, we will need to identify a 'bottleneck event' for each F ∈  .…”

“… 1 We remark that the range was of particular interest to the authors of [4] , who were motivated by an application to hypergraph Ramsey numbers, see [5] . …”

Small codes

“…Very recently, Conlon, Fox, He, Mubayi, Suk, and Verstraëte [5] established a connection between error‐correcting codes and a variant of the classical Ramsey numbers. For all $$q, r, s \in \mathbb {N}$$ with $$r > s$$, the set‐coloring Ramsey number $$R(q; r, s)$$ is defined to be the minimum $$n$$ such that if each edge of the complete graph $$K_n$$ receives $$s$$ colors out of a universe of $$r$$ colors, then there exist a set of $$q$$ vertices all of whose edges received the same color.…”

“…In particular, $$R(q; r, 1)$$ is just the usual $$r$$‐color Ramsey number of a $$q$$‐clique. For all $$q, r, s \in \mathbb {N}$$ with $$r > s$$, it was shown in [5] that $$$$\begin{equation} R(q+1; r, s) \geqslant A_q(r,s) + 1, \end{equation}$$$$and on the other hand, Conlon, Fox, Pham, and Zhao [6] showed that this bound is approximately tight when $$s$$ is near $$(1-1/q) r$$ by proving that for any $$\varepsilon > 0$$, there exists a $$c > 0$$ such that if $$r, s \in \mathbb {N}$$ with $$s \leqslant (1-1/q)r$$ and $$j = (1-1/q)r - s + 1$$, then …”