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Abstract: What is the largest number of projections onto k coordinates guaranteed in every family of m binary vectors of length n?

This fundamental question is intimately connected with important topics and results in combinatorics and computer science (Turan numbers, the Sauer-Perles-Shelah Lemma, the Kahn-Kalai-Linial Theorem), and is generally wide open.

We (essentially) settle the question for a wide range of parameters: linear k and sub-exponential m.

 Based on joint work with Noga Alon and Noam Solomon.