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Abstract: What is the largest number of projections onto k coordinates guaranteed in every set of m binary vectors of length n? This fundamental question is intimately connected to important topics and results in combinatorics and computer science (Turan numbers, the Sauer-Perles-Shelah Lemma, the Kahn-Kalai-Linial Theorem, and more), and is wide open for most settings of the parameters. We essentially settle the question for all linear k and sub-exponential m.

Based on joint work with Noga Alon and Noam Solomon