A note on natural correspondences that satisfy exclusion (Q2460079)

In this short paper of only 2 pages the author proves that if \(F: P_{2}(X) \rightarrow P(X)\) is a natural correspondence (i.e. for all \(S \in P_{2}(X)\), \(S \subseteq F(S)\)) that does not satisfy exclusion (i.e. for every \(T \in P_{3}(X)\) there is some \(S \in P_{2}(X)\) such that \(S \subseteq T \subseteq F(S)\) ), then the average image size \(a_F\) is greater than or equal to \(\frac{m+4}{3}\). Here \(m\) is the number of elements in \(X\) and \(P_{k}(X)\) is the collection of all \(k\)-element (\(k = 2,3\)) subsets of \(X\). In addition, this result is the best possible one.