Expanders graphs and sieving in combinatorial structures

TL;DR

This paper establishes a general large sieve framework for random walks on subgraphs, extending spectral gap results to combinatorial structures like colored integers and graphs, with high-probability expansion properties.

Contribution

It introduces a unified large sieve approach for combinatorial structures using expansion properties, generalizing spectral gap methods to new contexts.

Findings

Random colored subsets of integers contain monochromatic zero-sum subsets with high probability.

Random edge colorings of complete graphs contain monochromatic triangles with high probability.

The framework applies to various combinatorial problems involving expansion and sieving.

Abstract

We prove a general large sieve statement in the context of random walks on subgraphs of a given graph. This can be seen as a generalization of previously known results where one performs a random walk on a group enjoying a strong spectral gap property. In such a context the point is to exhibit a strong uniform expansion property for a suitable family of Cayley graphs on quotients. In our combinatorial approach, this is replaced by a result of Alon--Roichman about expanding properties of random Cayley graphs. Applying the general setting we show e.g., that with high probability (in a strong explicit sense) random coloured subsets of integers contain monochromatic (non-empty) subsets summing to zero, or that a random coloring of the edges of a complete graph contains a monochromatic triangle.