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Raghu Vardhan Reddy Meka: New Frontiers in Structure vs Randomness with Applications to Combinatorics, Complexity, Algorithms. #ICBS2024
In 1936, Erdos and Turan asked the following: Suppose you have a set S of integers from $\{1,2,..., N\}$ that contains at least $N / C$ elements. Then, for large enough N, must S have three equally spaced numbers (i.e., a 3-term arithmetic progression)? In 1946, Behrend showed that $C$ can be at most $exp(\Omega(\sqrt{\log N}))$. Since then, the problem has been a cornerstone of the area of additive combinatorics, with the best bound being $C = (\log N)^{1+c}$ for some constant $c (gt) 0$. Recent work obtained an exponential improvement showing that $C$ can be as big as $exp((\log N)^{0.09})$, thus getting closer to Behrend's construction. In this talk, I will describe this result and the main ingredient, a new variant of the "structure vs. randomness" paradigm and highlight two additional applications: 1. Communication complexity: explicit separations between randomized and deterministic multi-party protocols. 2. Algorithm design: fast combinatorial algorithms for detecting triangles in graphs.