On a conjecture of R. Rado

Abstract

Let (Ai i e{open) I) be an indexed family of nonempty intervals of a linearly ordered set (L, <). Let (I, E) be the intersection graph of (Ai i e{open) I), that is (i, j)e{open) E if and only if Ai∩ Aj ø. In [6], R. Rado considered the following sentence R(k), where K is a cardinal number. If Chr(J, E ∩ [J]2) ≤ K for all J ⊆ I, J ≤ K+, then Chr(I, E) ≤ k. He proved (see [6, Theorem 2]) that R(k) holds for every finite k, and conjectured (see [6, Conjecture 1]) that R(k) holds for every cardinal K. In this note we show that if R(N0) holds, then there is an inner model of set theory with many measurable cardinals. On the other hand, using consistency of the existence of a supercompact cardinal, we prove that R(N0) is consistent with the usual axioms of set theory. We also prove a few results about the intersection graph of (Ai i e{open) I).