Maximal Spread of Coherent Distributions: a Geometric and Combinatorial Perspective

We discuss some open problems concerning the maximal spread of coherent distributions. We prove a sharp bound on $\mathbb{E}|X-Y|^{\alpha}$ for $(X,Y)$ coherent and $\alpha \le 2$, and establish a novel connection between coherent distributions and such combinatorial objects as bipartite graphs, conjugate partitions and Ferrer diagrams. Our results may turn out to be helpful not only for probabilists, but also for graph theorists, especially for those interested in mathematical chemistry and the study of topological indices.