A counterexample to the Anstee-Sali's conjecture

Abstract

This note propose a counterexample to the Anstee--Sali conjecture for forbidden configurations. The basic candidate is the $4$-uniform family \[ F_2=\{xyab,xybc,xycd,xyda\} \] on six vertices: a fixed two-vertex core $\{x,y\}$ joined to the four edges of a $4$-cycle. We give an explicit certificate that every four-fold product whose factors are of type $I$, $I^c$, or $T$ contains $F_2$, while $I^3$ avoids it. Thus, $X(F_2)=4$, so the conjecture predicts $\operatorname{forb}(m,F_2)=Θ(m^3)$. On the other hand, a result of Mubayi on complete multipartite hypergraphs implies \[ \operatorname{forb}(m,F_2)=Ω(m^{7/2}), \] which is asymptotically larger than $m^3$. The example was found by GPT-5.6 Sol.

Disclosure

“4 AI-assistant disclosure The author used GPT-5.6 Sol to help identify the example and to assist with the exposition. The author independently reviewed and revised the outputs, verified the mathematical claims, and takes full responsibility for the final manuscript. References [1] R. P.”

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Classification

Suggesting mathematical examples or conjectures

Multiplier

6

Verified

2026-08-23T05:00:18.738420Z

Structural counts