A note on a set-mapping problem of Hajnal and Máté

Abstract

It is consistent that there exists a set mapping F with fl < F(fl, or) < a for )3 + 2 < a < w2 with no uncountable free sets.For our current purposes, a set mapping is a function F such that Dom(F) =[X] 2, Ran (F) _C [X] x (or C_ IX] <x) for some set X and cardinal A. Here, if X is a set, A a cardinal, [X] x = {Y C_C_ X: IYI = A}, [X] <x = {Y C_ X: IYI < A}. For a set mapping as above, a set Y C X is free ify ~/F(xl, x2) for any different Xl, z2, y E Y. For theorems and problems about free sets see [2] and [3]. In [3], A. Hajnal and A.Ms asked if it is consistent that there is a set mapping F : [w2] 2 --~ [w2] <~ such that for/3 < a < w2, F(13, a) is a subset of the ordinal interval (13, a) = {7:13 < 3' < a} and no uncountable free set exists. Here we prove, using models of AbrahamShelah [1] that such functions can consistently exist. The models of [1] were created to show the consistency of the following statement. There exists a family of closed, unbounded sets {Ea : a < w2) in wl such that the intersection of any uncountably many of them is finite.THEOREM 1. It is consistent that there ezists a set mapping F : [w2] 2 --* [w2] -<s~ with F(13, a) C_ (13, a) and with no uncountable free set.We give two proofs for the Theorem, both using models of [1]. Assume that for a < w2 a function ~o~ : a ---* wl is given such that ~1(i) is countable for every i < wl. Assume further that E~ C wl is a closed, unbounded set for a <w2. Put for fl < a <w2 7 E F(13, a) if13 < 7 < aand ~a(7) < min(E~ -[~a(13) + 1]).FIRST PROOF. Let V be a model of ZFC, fix ~a as above for a < w2. The first model of [1] extends V to a model V" containing closed, unbounded sets Ea(a < w2),

Hajnal–Máté graphs, Cohen reals, and disjoint‐type guessing

“…Let us mention a few. Komjáth, in his series of papers about HM graphs [7–9], showed that one can construct a triangle‐free HM graph just from the $$\lozenge$$ principle. From $${\lozenge ^+}$$, he constructed an HM graph with no special cycles , that is, cycles formed from two monotone paths.…”