The size-Ramsey number of 3-uniform tight paths

Jie Han,Shoham Letzter,Yoshiharu Kohayakawa,Olaf Parczyk,Guilherme Oliveira Mota

doi:10.19086/aic.24581

Abstract

Given a hypergraph H, the size-Ramsey number r(H) is the smallest integer m such that there exists a graph G with m edges with the property that in any colouring of the edges of G with two colours there is amonochromatic copy of H. We prove that the size-Ramsey number of the 3-uniform tight path on n vertices P_n is linear in n, i.e., r(P_n)=O(n). This answers a question by Dudek, Fleur, Mubayi, and Rödl for 3-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved r(P_n)=O(n^1.5*log^1.5 n).

Highlights

For hypergraphs G and H and an integer s, we denote by G → (H)s the property that in any s-colouring of the edges of G there is a monochromatic copy of H
The systematic investigation of size-Ramsey questions for hypergraphs was initiated by Dudek, La
La Fleur, Mubayi, and Rödl [8] asked if r2(Pn(r)) = Or(n) for r ≥ 3. We answer this question for 3-uniform hypergraphs by proving the following result

Summary

Introduction

For hypergraphs G and H and an integer s, we denote by G → (H)s the property that in any s-colouring of the edges of G there is a monochromatic copy of H. The systematic investigation of size-Ramsey questions for hypergraphs was initiated by Dudek, La. Fleur, Mubayi, and Rödl [8]. La Fleur, Mubayi, and Rödl [8] asked if r2(Pn(r)) = Or(n) for r ≥ 3 We answer this question for 3-uniform hypergraphs by proving the following result. The 2-colour size-Ramsey number of the 3-uniform tight path is r2(Pn(3)) = O(n). For any integer r such that 3 | r, the 2-colour size-Ramsey number of the r-uniform (2r/3)-path is r2(Pn(,r2)r/3) = O(n). The question whether the size-Ramsey number of a tight path is linear for hypergraphs with uniformity r ≥ 4 remains open and requires additional ideas

Sketch of the proof of Theorem 1

Sparse graphs with many long paths

Proof of Theorem 1