On the number of extreme measures with fixed marginals (Q549550)

Let \(X,Y\) be finite sets and \(G\) be a group acting on \(X\) and \(Y\). Given two \(G\)-invariant probability measures \(\mu_1\) and \(\mu_2\), with full supports on \(X\) and \(Y\), respectively, denote by \(K(\mu_1,\mu_2)\) the convex set of all \(G\)-invariant probability measures on \(X\times Y\) with marginals \(\mu_1\) and \(\mu_2\). The authors prove an exact (in order of growth, as the cardinalities of \(X\) and \(Y\) grow) upper estimate for the number of extreme points of \(K(\mu_1,\mu_2)\). This improves an estimate given by \textit{K. R. Parthasarathy} [Proc. Indian Acad. Sci., Math. Sci. 117, No. 4, 505--515 (2007; Zbl 1143.46038)]. Connections with combinatorics of graphs are also discussed.