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.