Note on a theorem of J. Folkman on transversals of infinite families with finitely many infinite members

JOIJRN.&L

OF COMBINATORIAL

THEORY,

Series

B 30. 100-102 !I981 I

Note Note on a Theorem of J. Folkman on Transversals of Infinite Families with Finitely Many Infinite Members FRANCOIS BRY Universitk Pierre et Marie Curie, U.E.R. 48, 4, place Jussieu, 75230 Paris Cedex 05, France Communicated

by the Editors

Received November 15, 1979 In this note we show by a simple direct proof that Folkman’s necessary and sufficient condition for an infinite family of sets with finitely many infinite members to have a transversal implies Woodall’s condition. A short proof of Folkman’s theorem results by combining with Woodall’s proof.

Several authors, Brualdi and Scrimger [ 11, Folkman [2] and Woodall [5], have given necessary and sufficient conditions for an infinite family of sets with finitely many infinite members to have a transversal, generalising the well-known theorems of Hall [ 3 ] and Jung and Rado [4 1. In each case the proof of necessity is straightforward. Woodall’s proof of his theorem is one page long, and he also establishes simply the equivalence between his conditions and those of Brualdi and Scrimger. On the other hand the proof of Folkman’s theorem in 121 is quite intricate. In this note we show by a simple direct proof that Folkman’s condition implies Woodall’s condition. A short proof of Folkman’s theorem results by combining with Woodall’s proof. Throughout this note A = (Aili E 1) is a family of subsets of a set E such that A i is finite for all i E I, with lb0 finite. For J E I we abbreviate lJ iEJA i to A(J).

THEOREM1 (J. Folkman 121). A has a transversal if and only if (1.1) for every non-negative integer n and pair J, J’ of sets such that J’ G J C I, A(J)\P(J’) is finite and (A(J)\P(J’)) < IJ\J’ ) - n there is a finite set J” 5 J’ such IA(L)l 2 ILI + n.

that

all

finite

sets

Let X be a subset of E. If the subfamily 100 D095-8956/81/01010@-03$02.00/0 Copyright c 198 I by Academic Press, Inc. All rights of reproduction in any form reserved.

L

with

J” c L c J’

satisfy

(,4,/i E I,) has a transversal

THEOREMOFJ.FOLKMAN

101

disjoint from X (i.e., by the theorem of M. Hall, if IA(J)wI !s (Jl for all finite J c Z,) we denote by c(X) the union of the finite subsets J’ of I, such that lA(J)\XI = jJ/; otherwise we set c(X) = 0. THEOREM

2 (D. R. Woodall

151). A has a transversal if and only if

(2.1) the subfamily (Aili E Z,) has a transversal theorem /A(J)] 2 (J( for allfinite J c I,);

(i.e.. by HalZ’s

(2.2) for all non-empty subsets K of Z\Z,, and all subsets X of A(K) such that /X/ < ]Kl we have A(K)CI(& A(c(X)).

Clearly (1.1)implies(2.1)(taken=OandJ=J’=L).Tc~showthat(1.1) implies (2.2) we need the following easy lemma contained in both [ 2 1 and

151. Let X be a finite subset of E such that the famiZy (Aili E I,) has a transversal disjoint from X. Then any finite union (and intersection) of finite sets J contained in I,, satisfying ]A(J)\X] = ]J] also has this property. Zt follows that any finite subset of c(X) is contained in a finite set L E c(X) such that IA(L)\X/ = 1L I. LEMMA.

Proof of (1) * (2.2). Suppose that for some K E Z\Z, and X E A(K) with ]X/ < lK1 we have A(K)\XsA(c(X)). Set J’ = c(X) and J = c(X) U K. We have A(J)\P(J’) = A(K)\A (c(X)) c w(c(X)) and .J\J’ = K (since c(X) c I,. K c Z\Z,). The hypothesis of (1.1) with n = /K / -- /X\p(c(X))] is satisfied: hence there is a finite J” contained in J’ such that /A(L)] > /L / -t n for all finite sets L with J” c L c J’. We have n > 1, and hence ]A(J”)] > ]J”/ + n implies J” f: 0: thus c(X) containing J” is also non-empty. By the lemma since J” contained in J’ = c(X) is finite there is a finite set L with J” EL c.Z’ such that 1A(L)\XI = /L /. Since IA(L)] > (L j + n also, it follows that

IKI + IL1= IX\p(J’)I + n + IL1< IXJA(J’)/ + lW)/ = lX\A(J’)]

+ ]A(L)n

X/ + iA(L)\X/


REFERENCES 1. R. A. BRUALDI AND E. B. SCRIMGER, Exchange Combinatorial

Theoq

5 (1968),

244-257.

systems.

matchings

and transversals.

J

2. J. FOLKMAN.

infinite families with finnely many infinite member%. .I 200-220, 3. M. HALL. Distinct representatives of subsets, Bull. Amer. Malh. Sot. 54 (1948). 922.-926. 4. R. RAI)O. Note on the transtinite case of Hall’s theorem on representatives, J. Lortdon .%larh. Sot. 42 (1967), 321-324. 5. D. R. WOO~ALI., Two results on infinite transversals, in “Combinatorics. Proc. it172 Oxford Combinatorial Conference” t D. J. A. Welsh and D. R. Woodall, Fds. J. pp. 34 I --350. Institute of Mathematics and its Applications. Southend-on-Sea. 1972. Combinntorial

Transversals Theory

of

9 (1970),