[4] With Micha Sharir he set up a framework which eventually led Guth and Katz to the solution of the Erdős distinct distances problem.
[1] Elekes started his mathematical work in combinatorial set theory, answering some questions posed by Erdős and Hajnal.
[1][6] His interest later switched to another favorite topic of Erdős, discrete geometry and geometric algorithm theory.
In 1986 he proved that if a deterministic polynomial algorithm computes a number V(K) for every convex body K in any Euclidean space given by a separation oracle such that V(K) always at least vol(K), the volume of K, then for every large enough dimension n, there is a convex body in the n-dimensional Euclidean space such that V(K)>20.99nvol(K).
[7] Then Nets Katz and Larry Guth used them to solve (apart from a factor of (log n) 1/2 ) the Erdős distinct distances problem, posed in 1946.