Ramsey numbers of 3-uniform loose paths and loose cycles

TL;DR

This paper determines exact 2-color Ramsey numbers for loose paths and cycles in 3-uniform hypergraphs, extending previous asymptotic results without using the Regularity Lemma.

Contribution

It provides exact formulas for Ramsey numbers involving loose paths and cycles in 3-uniform hypergraphs, generalizing prior asymptotic findings and answering an open question.

Findings

Exact formulas for Ramsey numbers of loose paths and cycles.

Extension of previous asymptotic results to exact values.

Positive resolution of a question by Gyás and Raeisi.

Abstract

Haxell et. al. [%P. Haxell, T. Luczak, Y. Peng, V. R\"{o}dl, A. %Ruci\'{n}ski, M. Simonovits, J. Skokan, The Ramsey number for hypergraph cycles I, J. Combin. Theory, Ser. A, 113 (2006), 67-83] proved that the 2-color Ramsey number of 3-uniform loose cycles on $2n$ vertices is asymptotically $\frac{5n}{2}$. Their proof is based on the method of 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 $n\geq m\geq 3$, $R(\mathcal{P}^3_n,\mathcal{P}^3_m)=R(\mathcal{P}^3_n,\mathcal{C}^3_m)=R(\mathcal{C}^3_n,\mathcal{C}^3_m)+1=2n+\lfloor\frac{m+1}{2}\rfloor$ and for $n>m\geq3$, $R(\mathcal{P}^3_m,\mathcal{C}^3_n)=2n+\lfloor\frac{m-1}{2}\rfloor$. These give a positive answer to a question of Gy\'{a}rf\'{a}s and Raeisi [The Ramsey number of loose triangles and quadrangles in hypergraphs, Electron. J. Combin. 19 (2012), #R30].