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- W3083711241 abstract "A bipartite graph $H = left(V_1, V_2; E right)$ with $|V_1| + |V_2| = n$ is semilinear if $V_i subseteq mathbb{R}^{d_i}$ for some $d_i$ and the edge relation $E$ consists of the pairs of points $(x_1, x_2) in V_1 times V_2$ satisfying a fixed Boolean combination of $s$ linear equalities and inequalities in $d_1 + d_2$ variables for some $s$. We show that for a fixed $k$, the number of edges in a $K_{k,k}$-free semilinear $H$ is almost linear in $n$, namely $|E| = O_{s,k,varepsilon}(n^{1+varepsilon})$ for any $varepsilon > 0$; and more generally, $|E| = O_{s,k,r,varepsilon}(n^{r-1 + varepsilon})$ for a $K_{k, ldots,k}$-free semilinear $r$-partite $r$-uniform hypergraph. As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis parallel sides in $mathbb{R}^d$ such that their incidence graph is $K_{k,k}$-free, there can be at most $O_{k,varepsilon}(n^{1+varepsilon})$ incidences. The same bound holds if instead of boxes one takes polytopes cut out by the translates of an arbitrary fixed finite set of halfspaces. We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in $o$-minimal structures (showing that the failure of an almost linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner)." @default.
- W3083711241 created "2020-09-11" @default.
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- W3083711241 date "2020-09-07" @default.
- W3083711241 modified "2023-09-27" @default.
- W3083711241 title "Zarankiewicz's problem for semilinear hypergraphs" @default.
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