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This is a crosspost from MathStackExchange (original question). Fix $k>0$ and let $X, Y$ be two vertex sets of size $n$ a positive integer (we're interested in the limit $n\to \infty$). Define a ...

This question is a modification of the one asked here, which turned out to ask for something too strong to be true. Given $k>0$ and a positive integer $n$, let $X, Y$ be two vertex sets of size $n$ ...