The range of thresholds for diameter 2 in random Cayley graphs

Given a group G , the model G ( G , p ) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p . a family of groups ( G k ) and a c ∈ R + we say that c is the threshold for diameter 2 for ( G k ) if for any ε > 0 with high probability Γ ∈ G ( G k , p ) has diameter greater than 2 if p ⩽ ( c − ε ) log n n and diameter at most 2 if p ⩾ ( c + ε ) log n n . In Christofides and Markström (in press)  [5] we proved that if c is a threshold for diameter 2 for a family of groups ( G k ) then c ∈ [ 1 / 4 , 2 ] and provided two families of groups with thresholds 1 / 4 and 2 respectively. s paper we study the question of whether every c ∈ [ 1 / 4 , 2 ] is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every c ∈ [ 1 / 4 , 4 / 3 ] is a threshold but a c ∈ ( 4 / 3 , 2 ] is a threshold if and only if it is of the form 4 n / ( 3 n − 1 ) for some positive integer n .