arXiv subjects

Hongyi Gou

Publications and source records attributed to Hongyi Gou.

2 recordsLinked to original sources

arXiv2026

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

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 $\varphi$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $\varphi$ on $n$ vertices has $O_{T,\varphi,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $\alpha_k=\min\{k-1,2\}$. 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)\alpha_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.

math.LO↗

arXiv2026

On Zarankiewicz's bounds for valued vector spaces

We establish absolute and relative almost-linear Zarankiewicz bounds for semilinear relations in valued vector spaces. For every fixed arity and description complexity, a $K_{t,\ldots,t}$-free semilinear $r$-partite hypergraph has at most \[ O\!\left(n^{r-1}(\log n)^c\right) \] edges, where $c$ depends only on the arity and the number of valuative literals. In the bipartite case a separate arbitrary-trace argument gives the explicit bound $O(n(\log n)^{2s})$ for description complexity $(\rho,s)$. We also prove a relative extension theorem: intersecting any relation with a hereditary almost-linear profile by $s$ affine moving-radius comparisons increases the logarithmic exponent by at most $2s$. For the additive affine-valuative structures on $\mathbb Q_p$ and $\mathbb C_p$, quantifier elimination converts these semilinear results into bounds for all definable relations. Finally, over every valued field with infinite value group, we construct $K_{2,2}$-free semilinear point--box graphs of description complexity $(1,4)$ with $\Omega(n\log n/\log\log n)$ edges.

math.LO↗