Abstract
We determine the asymptotics of the largest family of qualitatively 2-independentk-partitions of ann-set, for everyk>2. We generalize a Sperner-type theorem for 2-partite sets of Körner and Simonyi to thek-partite case. Both results have the feature that the corresponding trivial information-theoretic upper bound is tight. The results follow from a more general Sperner capacity theorem for a family of graphs in the sense of our previous work on Sperner theorems on directed graphs.
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References
Ahlswede, R. Coloring hypergraphs: A new approach to multi-user source coding. J. Comb. Inf. Syst. Sci., Pt. I,4, 76–115 (1979)
Berge, C.: Hypergraphes. Paris: Dunod 1988
Berge, C.: Graphs. 2nd revised edition. Amsterdam: North-Holland 1985
Bollobás, B.: On generalized graphs. Acta Math. Acad. Sci. Hungar.16, 441–452 (1965)
Cohen, G., Körner, J., Simonyi, G. Zero-error capacities and very different sequences. In: Sequences: combinatorics, compression, security and transmission. R.M. Capocelli ed., pp. 144–155 Springer-Verlag 1990
Csiszár, I., Körner, J.: Information Theory. Coding Theorems for Discrete Memoryless Systems. New York: Academic Press 1982
Feller, W.: An introduction to Probability Theory and its Applications. Vol. 1, 3rd edition. New York: Wiley 1968
Gargano, L., Körner, J., Vaccaro, U.: Sperner theorems on directed graphs and qualitative independence, J. Comb. Theory (A), to appear
Katona G.O.H.: Two applications (for search theory and truth functions) of Sperner type theorems, Periodica Math. Hung., 1–2(3), 19–26 (1973)
Körner, J.: Coding of an information source having ambiguous alphabet and the entropy of graphs, Transactions of the 6th Prague Conference on Information Theory, etc., 1971. pp. 411–425 Prague: Academia 1973
Körner, J., Simonyi G.: A Sperner-type theorem and qualitative independence. J. Comb. Theory (A)59 90–103 (1992)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math.13, 383–390 (1975)
Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. on Information Theory, IT-25, 1–7 (1979)
Poljak, S., Pultr, A., Rödl, V.: On the dimension of Kneser graphs. In: Algebraic Methods in Graph Theory. Szeged, Hungary, 1978, Coll. Soc. Math. J. Bolyai, pp. 631–646
Poljak, S., Pultr, A., Rödl, V.: On qualitatively independent partitions and related problems, Disc. Appl. Math.,6, 193–205 (1983)
Poljak, S., Rödl, V.: Orthogonal partitions and covering of graphs. Czech. Math. J.,30, pp. 475–485 (1980)
Poljak, S., Tuza, Z.: On the maximum number of qualitatively independent partitions. J. Comb. Theory (A)51, 111–116 (1989)
Rényi, A.: Foundations of Probability. New York: Wiley 1971
Shannon, C.E.: The zero-error capacity of a noisy channel. IRE Trans. on Information Theory,2, 8–19 (1956)
Sperner, E.: Ein Satz über Untermengen einer endlichen Menge. Math. Z.,27, 544–548 (1928)
Stein, S.K.: Two combinatorial covering theorems. J. Comb. Theory (A)16, 391–397 (1974)
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Work supported in part by the Italian Ministry of the University and of the Scientific Research in the framework of the “Algoritmi e Sistemi di Calcolo” project.
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Gargano, L., Körner, J. & Vaccaro, U. Sperner capacities. Graphs and Combinatorics 9, 31–46 (1993). https://doi.org/10.1007/BF01195325
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DOI: https://doi.org/10.1007/BF01195325