Analysis of a method coloring on study of Ramsey number R(3,10)

Abstract

Let s, t natural numbers; the Ramsey number R(s,t) is defined as the least positive integer $r$ with the property that every bicolored graph K_r contains one blue monocramatic subgraph K_s or one red monocramatic subgraph K_r. This theory gave rise to extensive research using, among other subjects, the study of combinatorics, started with Ramsey (1928). As simple as the definition is, calculating Ramsey numbers is very difficult and few are known. Exoo (1989), and Goedgebeur and Radziszowski (2013) showed that 40 <= R(3,10) <= 42. Thus, in this article, will be displayed studies and conclusions about R(3,10). We cannot yet state that the results presented in this article will be used in the final calculation of the Ramsey number R(3,10). The idea here is to share what we have studied in our research group, so that these studies can be used in the calculation of R(3,10) or to show colleagues who also study Ramsey numbers, which already we did, thus avoiding rework. Greenwood and Gleason (1955) used the notions of cubic and quadratic residues, respectively, to show that R(3,5)=14 and R(4,4)=17. Based on these ideas, given a complete graph with 41 vertices, in an isomorphic form, we will identify these vertices with the elements {0, ..., 40} of a field with 41 elements. And, with a bicoloration using residues of degree n module m (natural m, n), we will show that this graph contains a copy of blue K₃ or a red copy of K₁₀.