On explicit Ramsey graphs and estimates of the number of sums and products

In finite combinatorics there are many proofs of the existence of certain combinatorial structures which do not provide us with any explicit example of such structures. To give an explicit construction is not only a mathematical challenge, but often it is the only way to determine the extremal structures for a particular question. One of such problems is to give an explicit construction of a two-coloring of the complete bipartite graphKN,N such that no subgraphKr,r is monochromatic for some small r. It is well-known that there exist such colorings for r = (2 + o(1)) log2 N , but until recently explicit constructions were only known for r ≈ √ N . In 2004 Barak, Kindler, Shaltiel, Sudakov and Wigderson [1] found a polynomial construction of two-colorings of KN,N which leave no Kr,r monochromatic for r = N , where ε can be chosen arbitrarily small. Their result was a breakthrough not only in the field of Ramsey graphs, but they also succeeded in constructing extractors and other gadgets needed in derandomization with much better parameters. However, their construction is very complicated and uses derandomization. Thus it seems reasonable to look for more explicit constructions even if they have worse parameters. In this paper we give a very explicit construction of a three-coloring of KN,N in which no Kr,r is monochromatic for