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 $K_{t,\dots ,t}$-free semilinear $r$-partite hypergraph has at most $$O\text{\hspace{0.17em}⁣}(n^{r-1}(\log n{)}^c)$$ 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 $\mathrm{\Omega }(n\log n\mathrm{/}\log \log n)$ edges.