Graph with any rational density and no rich subsets of linear size
arxiv(2024)
摘要
A well-known application of the dependent random choice asserts that any
n-vertex graph G with positive edge density contains a `rich' vertex subset
U of size n^1-o(1) such that every pair of vertices in U has at least
n^1-o(1) common neighbors. In 2003, using a beautiful construction on
hypercube, Kostochka and Sudakov showed that this is tight: one cannot remove
the o(1) terms even if the edge density of G is 1/2. In this paper, we
generalize their result from pairs to tuples. To be precise, we show that given
every pair of positive integers p
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