The Hirsch Conjecture (1957) stated that the graph of a d-dimensional polytope with n facets cannot have (combinatorial) diameter greater than n − d. That is, any two vertices of the polytope can be connected by a path of at most n − d edges.This paper presents the first counterexample to the conjecture. Our polytope has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope with 48 facets that violates a certain generalization of the d-step conjecture of Klee and Walkup.

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Any configuration of lattice vectors gives rise to a hierarchy of higher-dimensional configurations which generalize the Lawrence construction in geometric combinatorics. We prove finiteness results for the Markov bases, Graver bases and face posets of these configurations, and we discuss applications to the statistical theory of log-linear models.Comment: 12 pages. Changes from v1 and v2: minor edits. This version is to appear in the Journal of Combinatorial Theory, Ser.

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