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Forbiddance and Capacity

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Abstract

We show that the maximum number of ternary sequences of length n such that no two of them feature all the three symbol pairs in their coordinates is 2(n+o(n)). In fact, we present a far more general theorem about problems of a similar nature. We explore the connections of our results to those on zero-error capacity of graph families.

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References

  1. Brightwell G., Cohen G., Fachini E., Fairthorne M., Körner J., Simonyi G., Tóth Á.: Permutation capacities of families of oriented infinite paths. SIAM J. Discrete Math. 2(24), 441–456 (2010)

    Article  Google Scholar 

  2. Calderbank R., Frankl P., Graham R.L., Li W., Shepp L.: The Sperner capacity of the cyclic triangle for linear and nonlinear codes. J. Algebraic Combin. 2, 31–48 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cohen G., Körner J., Simonyi G.: Zero-error capacities and very different sequences. In: Capocelli, R.M. (eds) Sequences: Combinatorics, Compression, Security and Transmission, pp. 144–155. Springer, New York (1988)

    Google Scholar 

  4. Csiszár I., Körner J.: On the capacity of the arbitrarily varying channel for maximum probability of error. Z. Wahrscheinlichkeitstheorie verw. Geb. 57, 87–101 (1981)

    Article  MATH  Google Scholar 

  5. Csiszár I., Körner J.: Information Theory: Coding Theorems for Discrete Memoryless Systems. Academic Press, New York (1982)

    Google Scholar 

  6. Diestel R.: Graph Theory, 3rd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  7. Ellis, D., Friedgut, E., Pilpel, H.: Intersection theorems for permutations (to appear)

  8. Erdős P., Ko C., Rado R.: Intersection theorems for systems of finite sets. Q. J. Math. Oxford Ser. 2 12, 313–320 (1961)

    Article  Google Scholar 

  9. Füredi Z., Kantor I., Monti A., Sinaimeri B.: On reverse free codes and permutations. SIAM J. Discrete Math. 3(24), 964–978 (2010)

    Article  Google Scholar 

  10. Gargano L., Körner J., Vaccaro U.: Sperner capacities. Graphs Comb. 9, 31–46 (1993)

    Article  MATH  Google Scholar 

  11. Gargano L., Körner J., Vaccaro U.: Capacities: from information theory to extremal set theory. J. Comb. Theory Ser. A 68(2), 296–316 (1994)

    Article  MATH  Google Scholar 

  12. Hahn, G., Tardif, C.: Graph homomorphisms: structure and symmetry. In: Graph Symmetry. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 497, pp. 107–166. Kluwer, Dordrecht (1997)

  13. Körner J., Orlitsky A.: Zero-error information theory. IEEE Trans. Inform. Theory 44, 2207–2229 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Körner J., Simonyi G.: A Sperner-type theorem and qualitative independence. J. Comb. Theory Ser. A 59, 90–103 (1992)

    Article  MATH  Google Scholar 

  15. Lovász L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25, 1–7 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  16. Marton K.: On the Shannon capacity of probabilistic graphs. J. Comb. Theory Ser. B 57, 183–195 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Nayak J., Rose K.: Graph capacities and zero-error transmission over compound channels. IEEE Trans. Inform. Theory 51(12), 4374–4378 (2005)

    Article  MathSciNet  Google Scholar 

  18. Scheinerman, E.R., Ullman, D.H.: Fractional Graph Theory. Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley, Chichester (1997)

  19. Shannon C.E.: The zero-error capacity of a noisy channel. IRE Trans. Inform. Theory 2, 8–19 (1956)

    Article  MathSciNet  Google Scholar 

  20. Simonyi G.: On Witsenhausen’s zero-error rate for multiple sources. IEEE Trans. Inform. Theory 49(12), 3258–3261 (2003)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, “La Sapienza” University of Rome, Via Salaria 113, 00198, Rome, Italy

    Emanuela Fachini & János Körner

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  1. Emanuela Fachini
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  2. János Körner
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Correspondence to János Körner.

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Fachini, E., Körner, J. Forbiddance and Capacity. Graphs and Combinatorics 27, 495–503 (2011). https://doi.org/10.1007/s00373-010-0987-9

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  • DOI: https://doi.org/10.1007/s00373-010-0987-9

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