ILL 1971 Problem 37

Let $S$ be a circle, and $\alpha =\{A_1,\ldots ,A_n\}$ a family of open arcs in $S$ . Let $N(\alpha )=n$ denote the number of elements in $\alpha$ . We say that $\alpha$ is a covering of $S$ if $\bigcup_{k=1}^n A_k\supset S$ . Let $\alpha=\{A_1,\ldots ,A_n\}$ and $\beta =\{B_1,\ldots ,B_m\}$ be two coverings of $S$ . Show that we can choose from the family of all sets $A_i\cap B_j,\ i=1,2,\ldots ,n,\ j=1, 2,\ldots ,m,$ a covering $\gamma$ of $S$ such that $N(\gamma )\le N(\alpha)+N(\beta)$ .