The Ramsey number of loose paths in 3-uniform hypergraphs

TL;DR

This paper determines the exact 2-color Ramsey number for 3-uniform loose paths when one path is significantly larger, extending previous asymptotic results to exact values in this specific case.

Contribution

It provides the exact 2-color Ramsey number for 3-uniform loose paths with one path much larger than the other, filling a gap in hypergraph Ramsey theory.

Findings

Exact formula for R(P^3_n, P^3_m) when n ≥ ⌊5m/4⌋

Extends asymptotic results to precise values for large paths

Identifies conditions under which the formula applies

Abstract

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$