On Zarankiewicz's bounds for valued vector spaces

Abstract

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 $(ρ,s)$. 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 $Ω(n\log n/\log\log n)$ edges.

Disclosure

“milinear of bounded complexity. Section 7 constructs the point–box examples giving the superlinear lower bound. Acknowledgements. The authors are grateful to Erik Walsberg and Artem Chernikov for helpful discus- sions related to this work. Artificial intelligence tools were used for writing and editing assistance in the preparation of this paper; the authors are responsible for the final content. 2. Preliminaries on valued vector spaces Let (K, vK ) be a valued field wit”

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2026-08-23T18:58:27.207315Z

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