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Will Sawin works on problems connecting arithmetic and combinatorics. His paper on the union-closed sets conjecture applies entropy inequalities to a finite family closed under unions.^[ref]

For a nontrivial such family, the paper guarantees an element in at least a $(3-\sqrt{5})/2$ fraction of the sets. The argument studies the union of independent random members and bounds the resulting entropy growth. It is a quantitative advance toward the conjectured one-half fraction, with the nonempty-union condition needed to exclude the family containing only the empty set.^[ref]