Combinatorics in 1-semi-equational theories

Abstract

We give direct combinatorial proofs that 1-semi-equational theories satisfy two combinatorial properties that imply the non-interpretability of certain fields. Our main result is that Boolean combinations of 1-semi-equations have almost linear Zarankiewicz bounds, and hence no infinite field is interpretable in a 1-semi-equational theory; this answers a question of Chernikov--Mennen and Chernikov--Starchenko, who had respectively established almost linear Zarankiewicz bounds for Boolean combinations of $(2,1)$-semi-equations and Boolean combinations of weakly normal relations. (This was recently and independently established by Gou, Mirabi, Mittal, Tran, and Yang, using different techniques and producing different bounds.) We also show that $(k,1)$-semi-equations satisfy the $\delta$-strong Erd\H{o}s--Hajnal property with $\delta=1/6^{k-1}$; this had previously been established by Chernikov--Starchenko for an ineffective constant $\delta>0$.