Hypergraph Ramsey numbers with quasipolynomial growth rate

Abstract

For a 3-uniform hypergraph (3-graph) $F$, let $r(F,n)$ be the smallest $N$ such that any $N$-vertex $F$-free 3-graph has an independent set of size $n$. We construct a $3$-graph $H_2$ with six vertices and five edges such that $r(H_2,n)=n^{Θ(\log n)}$, and a more general family of $3$-graphs $F$ for which $r(F,n)=n^{\log^{Θ(1)}(n)}$. These are the first examples of such Ramsey number known to be neither polynomial nor exponential.

Figures (4)

• Figure 1: An illustration of $H_2$.
• Figure 2: The sailboat graph $F$ with edges $abc$, $acd$, $ade$, $aef$, $afg$ and $bdg$ is another simple example with $r(F,n)=n^{\Theta(\log n)}$.
• Figure 3: Another illustration of $H_2$.
• Figure 4: An illustration of $H_3$. The red segments denote the link of $z$.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 1.2: Theorem 1.3 in conlon2024when
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5
• proof