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In this short note, we prove the following analog of theK\H{o}v\'ari-S\'os-Tur\'an theorem for intersection graphs of boxes. If $G$ isthe intersection graph of $n$ axis-parallel boxes in $\mathbb{R}^{d}$ such that$G$ contains no copy of $K_{t,t}$, then $G$ has at most $ctn(\log n)^{2d+3}$edges, where $c=c(d)>0$ only depends on $d$. Our proof is based on exploringconnections between boxicity, separation dimension and poset dimension.  Using this approach, we also show that a construction of Basit et al. of$K_{2,2}$-free incidence graphs of points and rectangles in the plane can beused to disprove a conjecture of Alon et al. We show that there exist graphs ofseparation dimension 4 having superlinear number of edges.

Tur\'an-type results for intersection graphs of boxes