The Zarankiewicz Problem for Polygon Visibility Graphs

We prove a quasi-linear upper bound on the size of $K_{t,t}$-free polygon visibility graphs. For visibility graphs of star-shaped and monotone polygons we show a linear bound. In the more general setting of $n$ points on a simple closed curve and visibility pseudo-segments, we provide an $O(n\log n)$ upper bound and an $\mathrm{\Omega }(n\alpha (n))$ lower bound.