Strategic Practice and Homework 8 - Projects at Harvard

We will prove the existence of a network with the desired property by showing thatthe probability is positive that a random network has the property is positive. Formarandomnetworkasin(a),andletA i be the event that the ith group of k people(in any fixed ordering) is neither a clique nor an anticlique. We havewhich shows that([n k)X( n k)P ( A c i) apple P (A c i)=i=1i=1i=1✓ nk◆2 (k 2)+1 < 1,(\n k)([n k)P ( A i )=1 P ( A c i) > 0,as desired. Alternatively, let C be the number of cliques of size k and A be thenumber of anticliques of size k, andwriteC + A = T .Then✓ nE(T )=E(C)+E(A) = 2k◆(k2)+1 < 1by the method of Part (a). So P (T =0)> 0, since P (T 1) = 1 would implyE(T ) 1. This again shows that there must be a network with the desired property.i=18