Lecture 10 : Random Ordering and Greedy Selection

for some constant C. We may want to colour every vertex uniformly at random. Then we will get m21−k many monochromatic edges in expectation. An improvement upon the basic method can be obtained by randomly recolouring all monochromatic edges after the first randomization. In a 1978 proof, Beck used this argument to show that m(k) ≥ Ω(k2). In 2000, Radhakrishnan and Srinivasan improved it to m(k) ≥ Ω((k/ log k)2). Their main idea is that the recolouring obeys some random ordering, and when we are about to recolour an edge, it may not be monochromatic any longer, and thus we do not need to recolour every monochromatic edge. Cherkashin and Kozik (2015) found a simpler proof based on a random greedy colouring, which implicitly used the recolouring idea. We will present that proof here.