The multicolour size Ramsey number of a path

TL;DR

This work resolves the dependence of the multicolour size Ramsey number of a path on the number of colours. For fixed $r\ge 2$ and $k\ge 100\log r$, the authors show $\widehat{R}_r(P_k)=\Theta((r^2 \log r)\,k)$ by developing a novel randomized edge-colouring scheme that avoids large monochromatic components and a bootstrapping mechanism anchored by a key Lemma guaranteeing dense $P_k$-free subgraphs. The method hinges on iterative extractions of dense $P_k$-free subgraphs, a careful balance of round-by-round color allocations, and balls-and-bins (Poisson-tail) analysis to prove the dense substructures exist with high probability. The resulting bound significantly tightens the understanding of colour-parameter influences on size Ramsey numbers for paths and introduces robust techniques (Vizing-type and star-type colorings) for final clean-up coloring.

Abstract

In this paper, we determine the $r$-colour size Ramsey number of the path $P_k$, up to constants. In particular, for every fixed $r \geq 2$ and $k \geq 100\log r$, we have \[ \widehat{R}_r(P_k)=Θ((r^2 \log r) \, k).\] Perhaps surprisingly, we do this by improving the lower bound on $\widehat{R}_r(P_k)$.

Figures (2)

• Figure 1: Round $i$ of the colouring process.
• Figure 2: The complete colouring process.

Theorems & Definitions (34)

• Theorem 1.1
• Lemma 2.0: Key lemma
• Lemma 2.1: Vizing-type colouring
• proof
• Lemma 2.2: Star-type colouring
• proof
• Lemma 3.1
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Lemma 4.0