发表机构
National University of Singapore; Wesleyan University; Nankai University（新加坡国立大学; 卫斯理安大学; 南开大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对1-半等式理论中可定义的多部超图，证明了几乎线性的扎兰凯维茨界，给出了二部情形下的边数上界，并以k-维分层索引集族的incidence估计完成证明。
AI 中文摘要
我们研究1-半等式理论中可定义的多部超图，并证明对每个固定的阶数r≥2，都存在几乎线性的扎兰凯维茨界。更准确地说，若T是1-半等式理论，那么对每个公式φ、每个t≥2和每个r≥2，存在常数c，使得由φ在n个顶点上定义的不含K_{t,…,t}的r部超图有O_{T,φ,t,r}(n^{r-1}(1+log(1+n))^c)条边。在二部情形下，当由m个(k,1)-半等式的布尔组合定义的图不含K_{t,t}时，其边数为O_{k,t,m}(n(1+log(1+n))^{(m-1)(k-1)})。特别地，由单个(k,1)-半等式或其否定定义的关系具有线性界。证明基于k-维分层索引集族的 incidence 估计。
英文摘要
We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $φ$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $φ$ on $n$ vertices has $O_{T,φ,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $α_k=\min\{k-1,2\}$. In the bipartite case, a Boolean combination of $m$ $(k,1)$-semi-equations has $ O_{k,t,m}\!\left( n(1+\log(1+n))^{(m-1)α_k} \right) $ edges whenever it is $K_{t,t}$-free. In particular, a relation defined by one $(k,1)$-semi-equation or its negation has a linear bound. The proofs combine incidence estimates for indexed set systems with low-crossing orderings of finite $k$-wise laminar families. Consequently, no $1$-semi-equational theory locally trace-defines an infinite domain.
Comments17 pages
