A counterexample to Sidorenko's conjecture
We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.
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Updates: Sharing AI progress on mathematics (amazing!); and my lecture plansCombinatorics and moreOct 7, 2026
has 35 vertices and 66 edges and also disproves the related forcing conjecture. Ryser’s covering conjecture. I discussed Ryser’s conjecture in my birthday post for Ron Aharoni, including his proof of the tripartite case. The new counterexamples are intersecting -partite, -uniform hypergraphs…