Background
Type: Article

Ramsey numbers of 3-uniform loose paths and loose cycles

Journal: Journal of Combinatorial Theory. Series A (00973165)Year: January 2014Volume: 121Issue: Pages: 64 - 73
Omidi G.R.Shahsiah M.a
Bronze • GreenDOI:10.1016/j.jcta.2013.09.003Language: English

Abstract

The 3-uniform loose cycle, denoted by Cn3, is the hypergraph with vertices v1,v2,.,v2n and n edges v1v2v3,v3v4v5,.,v2n-1v2nv1. Similarly, the 3-uniform loose path Pn3 is the hypergraph with vertices v1,v2,.,v2n+1 and n edges v1v2v3,v3v4v5,.,v2n-1v2nv2n+1. In 2006 Haxell et al. proved that the 2-color Ramsey number of 3-uniform loose cycles on 2. n vertices is asymptotically 5n2. Their proof is based on the method of the Regularity Lemma. Here, without using this method, we generalize their result by determining the exact values of 2-color Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs. More precisely, we prove that for every n ≥ m ≥ 3,. R(Pn3,Pm3)=R(Pn3,Cm3)=R(Cn3,Cm3)+1=2n+⌊m+12⌋, and for every n > m ≥ 3, R(Pm3,Cn3)=2n+⌊m-12⌋. This gives a positive answer to a recent question of Gyárfás and Raeisi. © 2013 Elsevier Inc.