Lecture 9: Probabilistic methods

Previously in class we proved that if we color the edges of K6 using blue or red color, then either there is a blue K3 or a red K3 as a subgraph. Here Kn is the complete graph (every pair of vertices are connected) on n vertices. We can generalize the above concept and ask, are there complete graphs for which any 2-coloring gives rise to either a blue Kk or a red Kl. It has been shown that there always exists n, s.t., any two coloring of Kn will have a monochromatic blue Kk or monochromatic red Kl. The smallest such number n is called the Ramsey number R(k, l). It has been a big open question to find out the bounds on R(k, l). We will use probabilistic method to give a lower bound on the diagonal Ramsey number R(k, k). Call an edge coloring of Kn good, if there are no monochromatic Kk’s. The idea would be to randomly color the edges of the graph Kn. If there is a positive probability (over the random coloring) that none of the Kk subgraphs are monochromatic red or blue, then there exist a coloring which is good. We color every edge either red or blue independently with probability 1/2. There are in total ( n k ) subgraphs Kk for a Kn.