For any c<1, k≥3 and t≥1 there exist k-uniform hypergraphs with maximum clique size >c|V| whose maximum cliques cannot be pierced by t vertices.
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Constant-degree polynomial samplers over F_2^m produce distributions at total variation distance 1-o(1) from Ber(1/3)^⊗n, with concrete bounds for d=1,2,3 and a supporting lemma that no degree-d polynomial has bias exactly 1/3.
citing papers explorer
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Piercing all maximum cliques in hypergraphs
For any c<1, k≥3 and t≥1 there exist k-uniform hypergraphs with maximum clique size >c|V| whose maximum cliques cannot be pierced by t vertices.
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On Sampling Lower Bounds for Polynomials
Constant-degree polynomial samplers over F_2^m produce distributions at total variation distance 1-o(1) from Ber(1/3)^⊗n, with concrete bounds for d=1,2,3 and a supporting lemma that no degree-d polynomial has bias exactly 1/3.