  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.
