Bounds on Ramsey Games via Alterations

TL;DR

This paper refines the alteration method for constructing $H$-free graphs in random graphs, showing that removing all edges in $H$-copies at suitable probabilities preserves independence number, and applies this to analyze online graph Ramsey games.

Contribution

It introduces a refined alteration technique that removes all edges in $H$-copies, improving analysis of $H$-free graphs and online Ramsey games.

Findings

Removing all edges in $H$-copies preserves independence number at certain probabilities.

The refined method simplifies analysis of online graph Ramsey games.

Demonstrates improved bounds and techniques for $H$-free graph construction.

Abstract

We present a refinement of the classical alteration method for constructing $H$-free graphs: for suitable edge-probabilities $p$, we show that removing all edges in $H$-copies of the binomial random graph $G_{n,p}$ does not significantly change the independence number. This differs from earlier alteration approaches of Erd\H{o}s and Krivelevich, who obtained similar guarantees by removing one edge from each $H$-copy (instead of all of them). We demonstrate the usefulness of our refined alternation method via two applications to online graph Ramsey games, where it enables easier analysis.