 With Komlós and Szemerédi he proved the "ct"2/log "t" upper bound for the Ramsey number "R"(3,"t").The corresponding lower bound was proved by Kim only in 1995, a result that earned him a Fulkerson Prize.With Chvátal, Newborn, and Szemerédi, Ajtai proved the crossing number inequality, that any drawing of a graph with "n" vertices and "m" edges, where , has at least crossings.