On the Erd\H{o}s-Purdy problem and the Zarankiewitz problem for semialgebraic graphs

Erd\H{o}s and Purdy, and later Agarwal and Sharir, conjectured that any set of $n$ points in $\mathbb R^{d}$ determine at most $Cn^{d/2}$ congruent $k$-simplices for even $d$. We obtain the first significant progress towards this conjecture, showing that this number is at most $C n^{3d/4}$ for $k<d$. As a consequence, we obtain an upper bound of $C n^{3d/4+2}$ for the number of similar $k$-simplices determined by $n$ points in $\mathbb R^d$, which improves the results of Agarwal, Apfelbaum, Purdy and Sharir. This problem is motivated by the problem of exact pattern matching. We also address Zarankiewicz-type questions of finding the maximum number of edges in semi-algebraic graphs with no $K_{u,u}$. Here, we improve the previous result of Fox, Pach, Sheffer, Suk, and Zahl, and Do for $d\le 4$, as well as for any $d$ and moderately large $u$. We get an improvement of their results for any $d$ and $u$ for unit-distance graphs, which was one of the main applications of their results. From a more general prospective, our results are proved using classical cutting techniques. In the recent years, we saw a great development of the polynomial partitioning method in incidence geometry that followed the breakthrough result by Guth and Katz. One consequence of that development is that the attention of the researchers in incidence geometry swayed in polynomial techniques. In this paper, we argue that there is a number of open problems where classical techniques work better.