Triangle Ramsey numbers of complete graphs

TL;DR

The paper studies triangle Ramsey numbers within the $F$-Ramsey framework and proves that for all sufficiently large $t$, $r_{K_3}(K_t)=\binom{r(K_t)}{3}$. The authors reduce the problem to showing that any $r$-chromatic graph in which every edge lies in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles, using a Turán-type bound combined with a suite of coloring tools for locally sparse and degenerate graphs, including probabilistic coloring and concentration inequalities. They extend the triangle-count bound from linear-sized graphs to all graphs, culminating in the exact asymptotic determination of the triangle Ramsey number in the large-$t$ regime and addressing a question posed by Spiro. The work also outlines conjectures and directions for generalizing to $r_{K_s}(K_t)$ and related hypergraph Ramsey problems, highlighting connections to Ramsey–Turán theory and hypergraph Turán-type invariants.

Abstract

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total.

Theorems & Definitions (36)

• Theorem 1.1: Chvátal EFRS78
• Conjecture 1.2
• Theorem 1.3
• Proposition 1.4
• Proposition 1.4
• Theorem 1.5
• proof : Proof of \ref{['prop:ramsey-critical']}
• Theorem 2.1: Turán
• Theorem 2.2
• proof