An infinite version of Ryser's inequality
The following conjecture of R. Aharoni (1984), generalizing Ryser's inequality, is known: If r is a natural number, \(H\neq \{\emptyset \}\) is a system of sets, and \(| A\cap B| =r\) for any two sets A,B in H, then H has the transversal property, i.e. it has a one-to-one choice function. The author proves the conjecture using the axiom of choice. Also the following assertion (in author's terminology ''claim'') is proved: The number of sets of type II is finite for \(\beta\) finite, and is at most \(\beta\) if \(\beta\) is infinite. Here a number \(A\in H\) is called of type I if \(A\supseteq X\) and of type II otherwise, \(X=A_ 0\cap A_ 1\), where \(A_ 0\) is a smallest member of H and \(A_ 1\) is a smallest member of \(H-\{A_ 0\}:\beta =| A_ 1-X|\).
scientific article; zbMATH DE number 3989349