Almost-linear Zarankiewicz bounds in $1$-semi-equational theories

@misc{gou2026almostlinear,
  title = {{Almost-linear Zarankiewicz bounds in $1$-semi-equational theories}},
  author = {Hongyi Gou and Mostafa Mirabi and Mihir Mittal and Chieu-Minh Tran and Zhenyu Yang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.25464},
  eprint = {2608.25464},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25464},
}

TY  - UNPB
TI  - Almost-linear Zarankiewicz bounds in $1$-semi-equational theories
AU  - Hongyi Gou
AU  - Mostafa Mirabi
AU  - Mihir Mittal
AU  - Chieu-Minh Tran
AU  - Zhenyu Yang
PY  - 2026
DA  - 2026/09/05
PB  - arXiv
DO  - 10.48550/arxiv.2608.25464
UR  - https://arxiv.org/abs/2608.25464
AB  - We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity r≥2. If T is a $1$-semi-equational theory, then, for every formula φ and fixed t,r≥2, there is a constant c such that each K_t,…,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.
ER  -