Capacities: From Information Theory to Extremal Set Theory

Abstract Generalizing the concept of zero-error capacity beyond its traditional links to any sort of information transmission we give an asymptotic solution to several hard problems in extremal set theory within a unified, formally information-theoretic framework. The results include the solution of far-reaching generalizations of Renyi's problem on qualitatively independent partitions.

[1]  Zsolt Tuza,et al.  On the maximum number of qualitatively independent partitions , 1989, J. Comb. Theory, Ser. A.

[2]  R. McEliece,et al.  The Lovasz bound and some generalizations , 1978 .

[3]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .

[4]  Rudolf Ahlswede,et al.  Coloring hypergraphs: A new approach to multi-user source coding, 1 , 1979 .

[5]  Luisa Gargano,et al.  Sperner capacities , 1993, Graphs Comb..

[6]  Alfréd Rényi,et al.  Foundations of Probability , 1971 .

[7]  R. A. Cuninghame-Green,et al.  Packing and Covering in Combinatorics , 1980 .

[8]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[9]  R. M. Capocelli.  Sequences: combinatorics, compression, security, and transmission , 1990 .

[10]  László Lovász,et al.  On the ratio of optimal integral and fractional covers , 1975, Discret. Math..

[11]  Gábor Simonyi,et al.  A Sperner-Type Theorem and Qualitative Independence , 1992, J. Comb. Theory, Ser. A.

[12]  I. Csiszár.  Information Theory , 1981 .

[13]  E. Gilbert.  A comparison of signalling alphabets , 1952 .

[14]  D. Blackwell,et al.  The Capacity of a Class of Channels , 1959 .

[15]  B. Bollobás.  On generalized graphs , 1965 .

[16]  M. Tsfasman,et al.  Modular curves, Shimura curves, and Goppa codes, better than Varshamov‐Gilbert bound , 1982 .

[17]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[18]  I. Csiszár,et al.  On the capacity of the arbitrarily varying channel for maximum probability of error , 1981 .

[19]  Gérard D. Cohen,et al.  Zero-error capacities and very different sequences , 1990 .

[20]  Willem H. Haemers,et al.  On Some Problems of Lovász Concerning the Shannon Capacity of a Graph , 1979, IEEE Trans. Inf. Theory.

[21]  Alexander Schrijver,et al.  A comparison of the Delsarte and Lovász bounds , 1979, IEEE Trans. Inf. Theory.

[22]  Robert J. McEliece,et al.  New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities , 1977, IEEE Trans. Inf. Theory.

[23]  L. A. Shepp,et al.  The Sperner Capacity of Linear and Nonlinear Codes for the Cyclic Triangle , 1993 .

[24]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[25]  by Arch. Rat. Mech. Anal. , 2022 .

[26]  Luisa Gargano,et al.  Qualitative Independence and Sperner Problems for Directed Graphs , 1992, J. Comb. Theory, Ser. A.