Pass to Expander Paradigm

ABSTRACT: The pass-to-expander paradigm is a very powerful and versatile tool in modern combinatorics. It allows one to reduce solving many questions about completely arbitrary graphs to solving them on (usually weak) expander graphs. We will illustrate several variants of this method by means of recent applications, which include major progress towards the classical Erdős-Gallai cycle decomposition conjecture, rainbow Turá,n numbers for all cycles (which found surprising applications to coding theory, additive combinatorics, and discrete geometry), and Graham's rearrangement conjecture.

This is based on joint works with Bedert, Kravitz, Montgomery, and Müyesser; Alon, Sauermann, Zakharov, and Zamir; and Montgomery.