Set Mappings, Partitions, and Chromatic Numbers

Hajnal, A.; Mate, Attila

doi:10.1016/s0049-237x(08)71956-6

Cited by 29 publications

(31 citation statements)

References 23 publications

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“…This situation is reminiscent of the one of the HajnalMáté graphs dened on ω 1 (in [8]). These graphs are also dened by means of a ladder system X α | α ∈ S λ ω , by joining α < β with an edge if α ∈ X β .…”

Section: Ladder Graphsmentioning

confidence: 87%

“…These graphs are also dened by means of a ladder system X α | α ∈ S λ ω , by joining α < β with an edge if α ∈ X β . It is proven in [8] that the diamond ω 1 implies that there is a HajnalMáté graph of chromatic number ℵ 1 , while MA ℵ 1 implies that all such graphs have chromatic number ℵ 0 . Yet, the situation with respect to the continuum hypothesis is clearer with the HajnalMáté graphs (see [1] and [2]): we know that it is consistent that CH holds and all of these graphs have countable chromatic number, but we do not know the impact of CH on the ladder graphs dened here.…”

Section: Ladder Graphsmentioning

confidence: 99%

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A note on the Engelking-Karłowicz theorem

Abraham

Yin

2008

Acta Math Hung

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We investigate the chromatic number of innite graphs whose denition is motivated by the theorem of Engelking and Karªowicz (in [3]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected i for some a ∈ X ∩ Y the order-type of a ∩ X is dierent from that of a ∩ Y .In addition to the chromatic number χ(G) of these graphs we study χ κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.

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Section: Ladder Graphsmentioning

confidence: 87%

Section: Ladder Graphsmentioning

confidence: 99%

A note on the Engelking-Karłowicz theorem

Abraham

Yin

2008

Acta Math Hung

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“…= 0 , that is the image /(a, /}) always lies above a. That this restriction may be relevant is shown by an example from F. Galvin (see Hajnal and Mate (1975) with a n^n /(a, B) = 0, yet / has no free set of 3 elements. However, imposing an intersection condition or a chain condition on the range of / ensures a large free set.…”

Section: Restricted Set Mappingsmentioning

confidence: 99%

Set mappings defined on pairs

Williams

1981

J Aust Math Soc A

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A set mapping on pairs over the set S is a function/ such that for each unordered pair a of elements of S,f(a) is a subset of 5 disjoint from a. A subset H of S is said to be free for/if x £ f{{y, z}) ' o r all x, y, z from H. In this paper, we investigate conditions imposed on the range of/which ensure that there is a large set free for/. For example, we show that if/is defined on a set of size K ++ with always \f(a)\ < K then/has a free set of size K + if the range of/satisfies the »c-chain condition, or if any two sets in the range of / have an intersection of size less than 9 for some 9 with 9 < K.

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“…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.…”

mentioning

confidence: 99%

“…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.…”

mentioning

confidence: 99%