We show that approximately counting the vertices of a bounded 0/1 polytope, presented as a system of rational linear inequalities, is, informally speaking, NP-hard. In particular, there is no FPRAS for this problem unless RP=NP. The proof is by a reduction from approximately counting homomorphisms from a given graph to a particular four-vertex graph. The main proof ideas were found using GPT-5.6 Sol Ultra.
We give a new proof of the fact that every 2-coloring of the positive integers contains a monochromatic Pythagorean triple. This result was originally proven by Heule, Kullmann, and Marek with SAT solvers in 2016, answering a decades-old question of Graham. This proof uses elementary techniques and requires neither len...
We show that a sequence of simplicial polytopes approximating a smooth convex body has primitive Betti numbers diverging from the upper bound given by the $g$-theorem. This complements a result of Adiprasito-Nevo-Samper, who proved the corresponding divergence from the lower bound. The proof uses a new lower bound for...
We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm...
We show that every lattice weak Minkowski summand of a smooth polytope combinatorially isomorphic to a product of simplices has a quadratic triangulation. This is achieved by identifying this class of polytopes with the class of Nakajima polytopes. We combine these results to give a new proof that two lattice weak Mink...
Juliana Curtis, Tuong Le, Chayim Lowen· 0 citations
We give a short proof that for a given planar set of terminals $P$ there is at most one locally minimal tree with prescribed directions at $P$ (i.e. two locally minimal trees cannot coincide in $B_\varepsilon(P)$). The new ingredient is a combinatorial result which says that a certain embedding of a bipartite cubic gra...
We continue the study of the Erdős-Dushnik-Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positio...
Paul E. Howard, E. Tachtsis· arXiv.org· 0 citations
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