Geodesic
A semi-algebraic version of Zarankiewicz's problem
Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1785-1810.

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A bipartite graph G is semi-algebraic in Rd if its vertices are represented by point sets P,Q⊂Rd and its edges are defined as pairs of points (p,q)∈P×Q that satisfy a Boolean combination of a fixed number of polynomial equations and inequalities in 2d coordinates. We show that for fixed k, the maximum number of edges in a Kk,k​-free semi-algebraic bipartite graph G=(P,Q,E) in R2 with ∣P∣=m and ∣Q∣=n is at most O((mn)2/3+m+n), and this bound is tight. In dimensions d≥3, we show that all such semi-algebraic graphs have at most C((mn)d+1d​+ε+m+n) edges, where here ε is an arbitrarily small constant and C=C(d,k,t,ε). This result is a far-reaching generalization of the classical Szemerédi-Trotter incidence theorem. The proof combines tools from several fields: VC-dimension and shatter functions, polynomial partitioning, and Hilbert polynomials.
DOI : 10.4171/jems/705
Classification : 05-XX, 52-XX
Keywords: Semi-algebraic graph, extremal graph theory, VC-dimension, polynomial partitioning, incidences 
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     title = {A semi-algebraic version of {Zarankiewicz's} problem},
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Jacob Fox; János Pach; Adam Sheffer; Andrew Suk; Joshua Zahl. A semi-algebraic version of Zarankiewicz's problem. Journal of the European Mathematical Society, Tome 19 (2017) no. 6, pp. 1785-1810. doi : 10.4171/jems/705. http://geodesic.mathdoc.fr/articles/10.4171/jems/705/

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