A note on almost disjoint families

Abstract

We give a short proof of the existence of arbitrarily large chro matic almost disjoint systems. With the same method we solve a problem of P. Szeptycki. One of the questions posed by P. Erdos and A. Hajnal in [2] was, do there exist arbitrarily large chromatic almost disjoint systems of countable sets? This was eventually proved by G. Elekes and G. Hoffmann in [1]. Later the result was extended to systems of sets of larger cardinality ([3]). These proofs applied a Baire category type argument in a topological setting. Here we give a short, direct proof of the Elekes-Hoffmann-Komjath theorem. With the same method we settle a problem of P. Szeptycki ([4]). He asked if it is consistent that for every family K of almost disjoint countable subsets of R there are sets Bo, B1,.... C R such that every H e KH has 1 ,> are infinite cardinals, then there exists an almost disjoint system Kt C [S] for some ISI = 2' with Chr (K) > i'. Proof. Set '1 = {f: a -+ r, a < s,+}. Clearly I (D = 2'. For some of the functions f E 1i we define H(f) c [4?] as follows. Let f: a -K i. If it is possible to find a set X C a of order type ,t, cofinal in a (and so cf(a) = cf(,u)) such that f is constant on X, then set f E 4I* and let H(f) = {ff13: /3 E X} for one such X. If no such set X can be found, make f V 4D* and leave H(f) undefined. Our system is K = {H(f): f E Lemma 1. KH is almost disjoint. Proof. Assume that IH(f ) n H(g)i = p Then H(f) = {f y: y E X} and H(g) = {gi6: 6 E Y} where f: a -* K, g 3 -s, X C a, Y C: are cofinal subsets of Received by the editors November 20, 2007. 2000 Mathematics Subject Classification. Primary 03E05.