A d-maximal subset of the n-letter Hamming cube that contains no unit ball has size at most roughly n^d, and related bounds show all finite d-maximal sets are bounded and finitely many up to isomorphism.
Note on Sunflowers
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abstract
A sunflower with p petals consists of p sets whose pairwise intersections are identical. The goal of the sunflower problem is to find the smallest r=r(p,k) such that any family of r^k distinct k-element sets contains a sunflower with p petals. Building upon a breakthrough of Alweiss, Lovett, Wu and Zhang from 2019, Rao proved that r=O(p log(pk)) suffices; this bound was reproved by Tao in 2020. In this short note we record that r=O(p log k) suffices, by using a minor variant of the probabilistic part of these recent proofs.
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Maximal sets of a given diameter in Hamming cubes
A d-maximal subset of the n-letter Hamming cube that contains no unit ball has size at most roughly n^d, and related bounds show all finite d-maximal sets are bounded and finitely many up to isomorphism.