European Journal of Mathematics

, Volume 3, Issue 2, pp 429–432 | Cite as

A note on the size of \(\mathscr {N}\)-free families

  • Ryan R. Martin
  • Shanise WalkerEmail author


The \(\mathscr {N}\) poset consists of four distinct sets WXYZ such that \(W\subset X, Y\subset X\), and \(Y\subset Z\) where W is not necessarily a subset of Z. A family \({{\mathscr {F}}}\), considered as a subposet of the n-dimensional Boolean lattice \(\mathscr {B}_n\), is \(\mathscr {N}\)-free if it does not contain \(\mathscr {N}\) as a subposet. Let \(\mathrm{La}(n, \mathscr {N})\) be the size of a largest \(\mathscr {N}\)-free family in \(\mathscr {B}_n\). Katona and Tarján proved that Open image in new window , where Open image in new window and Open image in new window is the size of a single-error-correcting code with constant weight \(k+1\). In this note, we prove for n even and \(k=n/2, \mathrm{La}(n, \mathscr {N}) \geqslant {n\atopwithdelims ()k}+A(n, 4, k)\), which improves the bound on \(\mathrm{La}(n, \mathscr {N})\) in the second order term for some values of n and should be an improvement for an infinite family of values of n, depending on the behavior of the function Open image in new window .


Forbidden subposets Error-correcting codes Extremal set theory 

Mathematics Subject Classification




We would like to extend our thanks to Kirsten Hogenson and Sung-Yell Song for providing helpful conversations.


  1. 1.
    Brouwer, A.E., Shearer, J.A., Sloane, N.J.A., Smith, W.D.: A new table of constant weight codes. IEEE Trans. Inform. Theory 36(6), 1334–1380 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Erdős, P.: On a lemma of Littlewood and Offord. Bull. Amer. Math. Soc. 51(12), 898–902 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Graham, R.L., Sloane, N.J.A.: Lower bounds for constant weight codes. IEEE Trans. Inform. Theory 26(1), 37–43 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Griggs, J.R., Katona, G.O.H.: No four subsets forming an \(N\). J. Combin. Theory Ser. A 115(4), 677–685 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Griggs, J.R., Li, W.-T.: Progress on poset-free families of subsets. In: Beveridge, A., et al. (eds.) Recent Trends in Combinatorics. The IMA Volumes in Mathematics and its Applications, vol. 159, pp. 317–338. Springer, Cham (2016)CrossRefGoogle Scholar
  6. 6.
    Katona, G.O.H., Tarján, T.G.: Extremal problems with excluded subgraphs in the \(n\)-cube. In: Borowiecki, M., Kennedy, J.W., Sysło, M.M. (eds.) Graph Theory. Lecture Notes in Mathematics, vol. 1018, pp. 84–93. Springer, Berlin (1983)Google Scholar
  7. 7.
    Sperner, E.: Ein satz über utermengen einer endlichen Menge. Math. Z. 27, 544–548 (1928)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIowa State UniversityAmesUSA

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