Erdős Matching Conjecture for almost perfect matchings

06/03/2022

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by   Dmitriy Kolupaev, et al.

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In 1965 Erdős asked, what is the largest size of a family of k-elements subsets of an n-element set that does not have a matching of size s+1? In this note, we improve upon a recent result of Frankl and resolve this problem for s>101k^3 and (s+1)k≤ n<(s+1)(k+1/100k).