On a Conjecture of R. Rado

Let (At | i e I) be an indexed family of nonempty intervals of a linearly ordered set {L, <). Let (/, E) be the intersection graph of (̂ 4, | I G / ) , that is {i,j}eE if and only if At n At J= 0 . In [6], R. Rado considered the following sentence R(K), where K is a cardinal number. If Chr(J, E n |V]) ^ K for all ; £ / , |J| ^ K + , then Chr(/ , 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 /?(X0) 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 /?(X0) is consistent with the usual axioms of set theory. We also prove a few results about the intersection graph of (/I, | i e /). 1. Notation and definitions Let (L, < ) be a linearly ordered set. An interval of L is a nonempty subset A of L such that if x < y < z and x,z e A, then ye A. Let (A{ | i e I) be a family of intervals of L. The intersection graph of (At \ie I) is the graph (/, E), where E = {{ij} E [ /] | A, n Aj ± 0}. If ^ = (V, F) is a graph, then by Chr (^) we denote the chromatic number of ^ , that is the least cardinal K such that there is a representation " U vt, where [F{] 2 n F = 0 for all t, < K. If X is a cardinal, then JTA denotes the complete graph on X vertices. Let P be a partially ordered set, let a be an order type and let K be a cardinal. The symbol P -> (a),J means that for every partition P = [j P^, some P{ contains a chain of order type a. i<K A tree 7 is a partially ordered set such that {s e T | s < T t} is well-ordered under < T, for every t e 7. Thus T = (J Ta, where Ta, the a-th level of T, is the set of all as On t G T such that {s | s <Tt} has order type a. The height of T is the least a with Ta = 0. We say that T is K-special if and only if T is the union of ^ K antichains; T is special if and only if T is K0-special. Note that T Is K-special if and only if T + {K)1 Let P be a partially ordered set. Then by aP we denote the set of all bounded well-ordered subsets of P partially ordered by ^ , where s ^ t if and only if s is an intial segment of t. Then {aP, ^ ) is a tree. By o'P we denote the set {teaP\t has a maximal element} under the ordering ^ . All undefined terms can be found in any standard text of set theory. The results of this note were proved during the Symposium on Ordered sets in Banff (August-September, 1981). I should like to thank the organizing committee of the Symposium for making possible my attendance. Received 3 December, 1981. J. London Math. Soc. (2), 27 (1983), 1-8