Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers

TL;DR

This paper introduces an improved recursive bound for hypergraph Ramsey numbers using vertex online Ramsey numbers, leading to tighter upper bounds and new insights into hypergraph Ramsey theory.

Contribution

The authors generalize vertex online Ramsey numbers to hypergraphs and establish an improved recurrence relation, enhancing bounds on hypergraph Ramsey numbers.

Findings

Derived an improved recurrence relation for hypergraph vertex online Ramsey numbers.

Achieved a lower-order improvement to existing upper bounds for hypergraph Ramsey numbers.

Provided a refinement of the classical Erd ext{"o}s-Rado recursive bound.

Abstract

The classical recursive upper bound on hypergraph Ramsey numbers due to Erd\H{o}s and Rado states that for $2 \leq k < s \leq t$, \[ r_k(s,t) \leq 2^{\binom{r_{k-1}(s-1,t-1)}{k-1}}. \] In 2010, Conlon, Fox, and Sudakov introduced the so-called vertex online Ramsey numbers $\tilde{r}(s,t)$ for graphs to obtain a quantitative improvement over this bound when $k=3$. In this note, we show that the natural hypergraph generalization $\tilde{r}_k(s,t)$ of the vertex online Ramsey numbers satisfy an improved recurrence \[ \tilde{r}_k(s,t) \leq 2^{(1+o(1))\tilde{r}_{k-1}(s-1,t-1)}. \] We obtain several corollaries from this, including a lower-order improvement to the best known quantitative upper bounds for hypergraph Ramsey numbers and an improvement to the above recursive bound of Erd\H{o}s and Rado.