Abstract
AbstractAn old conjecture of Erdős and McKay states that if all homogeneous sets in an $n$ -vertex graph are of order $O(\!\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ . We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\!\log n)$ , then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega \big(n^2\big)\}$ .
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