An ABC conjecture proved

Not THE famous ABC conjecture, but a conjecture of mine with Mohammed Aljohani and John Bamberg. A simple counting argument shows that, in any vertex-transitive graph, the product of the clique number and the independence number is at most the number of vertices. This has extra significance because non-trivial examples meeting the bound are the obstructions to a permutation group property which we call separation arising in the theory of synchronizing automata. This puts strong restrictions on separating permutation groups, which we assume must be transitive: they must be primitive (else the disjoinot union of complete graphs on the blocks has product of clique and independence number equal to number of vertices); they must be basic (else they preserve a Hamming graph, which has this property); and so on … An interesting test case is formed by the symmetric groups Sn acting on the set of k-subsets of {1,…n}. (Without loss of generality we may assume that k≤n/2.) Any graph admitting…