Counterexamples to Sidorenko's conjecture and the forcing conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Counterexamples to Sidorenko's conjecture and the forcing conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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Counterexamples to Sidorenko’s conjecture and the forcing conjecture. Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.

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LEVEL 1 A counterexample to Sidorenko's conjecture

released 2026-09-23 | 1 theorem · 21 lemmas · 30 proofs · 21,832 words | PLAY LEVEL 1 » (pdf)

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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