The limit in the (k+2, k)-Problem of Brown, Erdős and Sós exists for all k≥ 2

2022/10/03 by Delcourt, Michelle, Postle, Luke
#05B07 #05C65 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.01105

Abstract

Let f^(r)(n;s,k) be the maximum number of edges of an r-uniform hypergraph on~n vertices not containing a subgraph with k~edges and at most s~vertices. In 1973, Brown, Erdős and Sós conjectured that the limit lim_{n→ ∞} n^-2 f⁽³⁾(n;k+2,k) exists for all positive integers k≥ 2. They proved this for k=2. In 2019, Glock proved this for k=3 and determined the limit. Quite recently, Glock, Joos, Kim, Kühn, Lichev and Pikhurko proved this for k=4 and determined the limit; we combine their work with a new reduction to fully resolve the conjecture by proving that indeed the limit exists for all positive integers k≥ 2.

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