On the Prague Dimension of Kneser Graphs
In this note we point out another connection between the Prague dimension of graphs and the dimension theory of partially ordered sets by giving a very short proof of a theorem of Poljak, Pultr and Rödl . We show that the dimension of the Kneser graph is bounded as dim P(K(n, k)) < C k log log n, where C k is depending only on k.
KeywordsLinear Extension Dimension Theory Boolean Lattice National Security Agency Kneser Graph
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- Z. Füredi, P. Hajnal, V. Rödl, and W. T. Trotter, “Interval orders and shift graphs”, Sets, Graphs and Numbers, A. Hajnal and V. T. Sós, Eds., Proc. Colloq. Math. Soc. Jcínos Bolyai 60, 297–313, (Budapest, Hungary, 1991), North-Holland, Amsterdam 1992, 297–313.Google Scholar
- L. Gargano, J. Körner and U. Vaccaro, “Capacity and dimension”, Lecture by J. Körner, Symposium Numbers, Information and Complexity in honor of R. Ahlswede, Bielefeld, Germany, October 1998.Google Scholar
- J. Körner and A. Monti, “Compact representations of the intersection structure of families of finite sets”, manuscript, November 1998.Google Scholar
- J. Körner and A. Orlitzky, “Zero-error information theory”, IEEE Trans. Information Theory, 50’th anniversary volume, to appear.Google Scholar
- J. Nesetril and A. Pultr, “A Dushnik-Miller type dimension of graphs and its complexity”, Fundamentals of Computation Theory, Proc. Conf. Poznan—Kórnik, 1977, Springer Lect. Notes in Comp. Sci. 56, 1977, 48 2493.Google Scholar
- S. Poljak, A. Pultr and V. Rödl, “On the dimension of Kneser graphs”, Algebraic Methods in Graph Theory, Proc. Colloq. in Szeged, Hungary, 1978, L. Lovâsz and V. T. Sós, Eds., Proc. Colloq. Math. Soc. J. Bolyai 25, 1981, 631–646.Google Scholar
- W.T. Trotter, Cornbinatorics and Partially Ordered Sets: Dimension Theory, John Hopkins University Press, Baltimore, Maryland, 1991. Also: “Progress and new directions in dimension theory for finite partially ordered sets”, Extremal Problems for Finite Sets, Proc. Colloq., Visegrâd, Hungary, 1991, P. Frankl et al., Eds., Bolyai Soc. Math. Studies 3, 1994, 457–477.Google Scholar