New lower bounds on the size-Ramsey number of a path

TL;DR

This paper establishes new lower bounds on the size-Ramsey number of paths, demonstrating that graphs with fewer edges than previously known can be 2-colored to avoid long monochromatic paths, and extends these bounds to multiple colors.

Contribution

The paper introduces improved lower bounds on the size-Ramsey number of paths for both 2-color and multi-color cases, advancing understanding of graph colorings and monochromatic path lengths.

Findings

Graphs with up to (3.75 - o(1))n edges can be 2-colored to avoid long monochromatic paths.

Previous bounds were up to 2.5n - 7.5 edges, now improved.

Enhanced lower bounds for the r-color case.

Abstract

We prove that for all graphs with at most $(3.75-o(1))n$ edges there exists a 2-coloring of the edges such that every monochromatic path has order less than $n$. This was previously known to be true for graphs with at most $2.5n-7.5$ edges. We also improve on the best-known lower bounds in the $r$-color case.