Counterexample to Ziegler's conjecture: explicit simplicial 7-dimensional 0/1-polytope with 14 vertices that is not centrally symmetric, plus enumeration of all such examples in dimension 7.
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Random 0/1-polytopes sampled with constant probability p have edge-expansion Θ(n) for p > 1/2 and n^Θ(log log n) for p < 1/2 with high probability.
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A Counterexample to Ziegler's Cross-Polytope Conjecture for Simplicial 0/1-Polytopes
Counterexample to Ziegler's conjecture: explicit simplicial 7-dimensional 0/1-polytope with 14 vertices that is not centrally symmetric, plus enumeration of all such examples in dimension 7.
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Random 0/1-polytopes expand rapidly
Random 0/1-polytopes sampled with constant probability p have edge-expansion Θ(n) for p > 1/2 and n^Θ(log log n) for p < 1/2 with high probability.