Roth’s theorem is a classical result in additive combinatorics, stating that every sufficiently large dense set of integers contains a three-term arithmetic progression. Although the theorem is arithmetic in nature, one of its most elegant proofs comes from graph theory.

Szemerédi’s Regularity Lemma reveals a remarkable structural property of large graphs: every sufficiently large graph can be decomposed into a bounded number of parts that behave in an approximately random way. Together with related counting ideas, this creates a powerful bridge between graph theory and additive combinatorics.

In this talk, I will introduce Roth’s theorem and the basic graph-theoretic notions needed to understand regularity, explain the main ideas behind Szemerédi’s Regularity Lemma and the Counting Lemma, and then show how these tools lead to a graph-theoretic proof of Roth’s theorem.

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Happy Holidays!

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