Ramsey numbers of connected 4-clique matchings

TL;DR

The paper determines the exact 2-color Ramsey number for connected $nK_4$ matchings as $R_2(c(nK_4))=13n-3$ for all $n\ge3$, with the $n=2$ case conditional on $R_2(2K_4)\le23$. It extends Roberts' previous result, which required $n\ge R(4,4)=18$, to all $n\ge3$ by developing a partition-based method and a red $K_2$ counting framework via the function $f_k(u)$ to force a red $nK_4$ under any red-blue coloring of $K_{13n-3}$. The key technique decomposes blue $K_4$ copies into either three parts of size at least $\lceil n/4\rceil$ or two parts of size at least $n-2$, and leverages these structures to assemble a red $nK_4$; the result for $n=2$ relies on a known bound $R_2(2K_4)\le23$. This work sharpens the understanding of Ramsey numbers for connected clique matchings and highlights remaining questions about small-n behavior, especially for $r\ge5$.

Abstract

In this paper, we determine the exact value of the $2$-edge-coloring Ramsey number of a connected $4$-clique matching $c(nK_4)$, which is a set of connected graphs containing an $nK_4$ is $13n-3$ for any positive integer $n \geq 3$. This is an extension of the result by Roberts (2017), which is proved only for $n\geq 18$. We also show that the result still holds when $n=2$ provided that $R_2(2K_4) \leq 23$.

Theorems & Definitions (9)

• Theorem 1
• Lemma 2
• proof
• Theorem 3
• Lemma 4
• proof
• Theorem 5
• Theorem 6
• proof