发表机构
Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences（纯数学与数学统计系，数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过直接组合证明，表明1-半等式理论的布尔组合具有几乎线性Zarankiewicz界，从而排除无限域的可解释性，并证明$(k,1)$-半等式满足强Erdős--Hajnal性质。
AI 中文摘要
我们给出了直接的组合证明，表明1-半等式理论满足两个组合性质，这些性质蕴含某些域不可解释性。我们的主要结果是，1-半等式的布尔组合具有几乎线性的Zarankiewicz界，因此任何无限域都不能在1-半等式理论中被解释；这回答了Chernikov--Mennen和Chernikov--Starchenko的问题，他们分别建立了$(2,1)$-半等式的布尔组合和弱正规关系的布尔组合的几乎线性Zarankiewicz界。（这一结果最近由Gou、Mirabi、Mittal、Tran和Yang独立获得，他们使用了不同的技术并产生了不同的界。）我们还证明了$(k,1)$-半等式满足$\delta$-强Erdős--Hajnal性质，其中$\delta=1/6^{k-1}$；这之前由Chernikov--Starchenko以无效常数$\delta>0$建立。
英文摘要
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 $δ$-strong Erdős--Hajnal property with $δ=1/6^{k-1}$; this had previously been established by Chernikov--Starchenko for an ineffective constant $δ>0$.
Comments14 pages; minor presentation and citation changes
