Ramsey games with giants

TL;DR

This paper explores a Ramsey-type variant of the giant component emergence in random graphs, analyzing multiple settings and providing bounds that are sometimes asymptotically tight, advancing understanding of colored graph processes.

Contribution

It introduces and analyzes various online and offline Ramsey-style problems related to giant components in colored random graphs, offering new bounds and insights.

Findings

Bounds for various problem variants are established.

In some cases, bounds are asymptotically tight.

The work extends classical random graph results to colored, multi-player scenarios.

Abstract

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.