Injective choice functions for countable families

Podewski, Klaus-Peter; Steffens, K.

doi:10.1016/0095-8956(76)90025-3

Cited by 40 publications

(22 citation statements)

References 9 publications

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“…The proof took more than forty years to discover. The first progress was by Podewski and Steffens, who proved König's duality theorem for countable bipartite graphs [7]. Aharoni next proved König's duality theorem for arbitrary bipartite graphs [1].…”

Section: Menger's Theorem In Every Web (G a B) There Is A Set Of Dmentioning

confidence: 99%

Menger’s theorem in $${{\Pi^1_1\tt{-CA}_0}}$$

Shafer

2012

Arch. Math. Logic

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Section: Menger's Theorem In Every Web (G a B) There Is A Set Of Dmentioning

confidence: 99%

Menger’s theorem in $${{\Pi^1_1\tt{-CA}_0}}$$

Shafer

2012

Arch. Math. Logic

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“…297ff.). Damerell and Milner (1974) gave a criterion for deciding whether a countable family of sets has a transversal; an alternative criterion was given by Podewski and Steffens (1976) and Nash-Williams (1978). Shelah (1973) provided an inductive criterion which together with the other results resolved the issue for the case of countable collections of countable sets.…”

Section: Introductionmentioning

confidence: 99%

Transversals, systems of distinct representatives, mechanism design, and matching

Hurwicz¹,

Reiter²

2001

Rev Econ Design

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A transversal generated by a system of distinct representatives (SDR) for a collection of sets consists of an element from each set (its representative) such that the representative uniquely identifies the set it belongs to. Theorem 1 gives a necessary and sufficient condition that an arbitrary collection, finite or infinite, of sets, finite or infinite, have an SDR. The proof is direct, short. A Corollary to Theorem 1 shows explicitly the application to matching problems. In the context of designing decentralized economic mechanisms, it turned out to be important to know when one can construct an SDR for a collection of sets that cover the parameter space characterizing a finite number of economic agents. The condition of Theorem 1 is readily verifiable in that economic context. Theorems 2-5 give different characterizations of situations in which the collection of sets is a partition. This is of interest because partitions have special properties of informational efficiency.

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“…The converse of the Theorem holds for countable families. Let G be a maximal critical subfamily of a countable family F. Then by Theorem 8 of [5], Lemma 3 and Lemma 4, (F\kernel F) u G is a maximal representable subfamily of F.…”

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confidence: 97%

“…If F has a maximal representable subfamily M, then by the Theorem there is a maximal critical subfamily G £ F such that M = (F\kernel F) u G. By Lemma 3, L4(F kerneI F ) # 0 . Conversely, suppose that F kerncI F has an i.c.f.. Lemma 1 of [5] implies that F has a maximal critical subfamily G. Let M = (F\kernel F) u G, then M has an i.c.f. To prove that M is maximal, let i e dmn (F\M).…”

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confidence: 99%