Hodge originally formulated his conjecture with integral coefficients. Atiyah and Hirzebruch showed in 1962 that it fails for torsion classes, and later Kollár produced non-torsion counterexamples. Since then, the conjecture has been understood with rational coefficients, while its integral form is regarded as a property of individual varieties rather than a universal statement. A longstanding open case concerned the integral Hodge conjecture for abelian varieties. In this talk, I will explain how ideas from the combinatorial theory of regular matroids allow us to prove that it fails for large classes of abelian varieties. As a consequence, building on work of Voisin, we show that very general cubic threefolds are not stably rational, strengthening the classical result of Clemens and Griffiths (1972) that smooth cubic threefolds are not rational. This is joint work with Philip Engel and Olivier de Gaay Fortman.