A counter-example to the probabilistic universal graph conjecture via randomized communication complexity

11/19/2021

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by   Lianna Hambardzumyan, et al.

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We refute the Probabilistic Universal Graph Conjecture of Harms, Wild, and Zamaraev, which states that a hereditary graph property admits a constant-size probabilistic universal graph if and only if it is stable and has at most factorial speed. Our counter-example follows from the existence of a sequence of n × n Boolean matrices M_n, such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every √(n)×√(n) submatrix of M_n is bounded by a universal constant.