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

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 Mn, such that their public-coin randomized communication complexity tends to infinity, while the randomized communication complexity of every √ n × √n submatrix of Mn is bounded by a universal constant.