Consider a complete graph Kn with n nodes. That is, a graph with nodes lG={1. .,n}, and all possible (3) edges i.e. all pairs of nodes {Lj} C G are connected with an edge. Let C( (n ,m)= (m ). Show that for any integer with k o (2) 31:1 and conclude that therefore there is at least one graph Kn with *no* size is: subgraph of single-color. Which of the following arguments proves the claim of equation (2]? Also see Section 6.1 in the book for more discussion. However, Section 6.1. is not strictly needed. The problem uses only methods from Section 1. If you just follow the hint and instructions in the HW problem, that's enough. OPr(E,:)- (1Pr(E,))=HPr(E,) >0 :2 (2) 0 Pas): (3)2 0M\" Pr(E,;) > 0 _> (2) O Pr (Ei) = 2 0&2\" 1 for coloring edge 2, . . . , C" (k. 2) the same color as the 1st edge in the i-th subgraph. Then P1'(NSG) : Pr (r1521 Bi) 2 1 2P1\" (Eg } 1 (k)26',[k2}+1>0 by eq. (1). Therefore there is at least one outcome, i.e., graph with (2] 0 By Markov inequality Pr (C (mk) > 1) 2 m = (E) 2_c[k'2)+1 > U by eq. (1)- 0 none of these