, Volume 31, Issue 3, pp 421–433 | Cite as

Diamond-Free Subsets in the Linear Lattices

  • Ghassan Sarkis
  • Shahriar Shahriari


Four distinct elements a, b, c, and d of a poset form a diamond if \(a< b<d\) and \(a<c<d\). A subset of a poset is diamond-free if no four elements of the subset form a diamond. Even in the Boolean lattices, finding the size of the largest diamond-free subset remains an open problem. In this paper, we consider the linear lattices—poset of subspaces of a finite dimensional vector space over a finite field of order q—and extend the results of Griggs et al. (J. Combin. Theory Ser. A 119(2):310–322, 2012) on the Boolean lattices, to prove that the number of elements of a diamond-free subset of a linear lattice can be no larger than \(2+\frac {1}{q+1}\) times the width of the lattice, so that this fraction tends to 2 as \(q \longrightarrow \infty \). In addition, using an algebraic technique, we introduce so-called diamond matchings, and prove that for linear lattices of dimensions up to 5, the size of a largest diamond-free subset is equal to the sum of the largest two rank numbers of the lattice.


Diamond-free Linear lattices Subspace lattices Lubell function Forbidden poset 

Mathematics Subject Classifications (2010)

Primary 06A07; Secondary 05D05 05D15 


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  1. 1.
    Axenovich, M., Manske, J., Martin, R.: Q 2-free families in the Boolean lattice. Order 29, 177–191 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Balogh, J., Hu, P., Lidický, B., Liu, H.: Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube. arXiv:1201.0209v2 (2012)
  3. 3.
    Bukh, B.: Set families with a forbidden subposet. Electron. J. Combin. 16(1) (2009). Research Paper 142, 11. MR 2577310 (2011b:06005)Google Scholar
  4. 4.
    Carroll, T., Katona, G.O.H.: Bounds on maximal families of sets not containing three sets with \(A\cap B\subseteq C,\ A\nsubseteq B\). Order 25(3), 229–236 (2008). MR 2448406 (2009k:05185)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Bonis, A., Katona, G.O.H.: Largest families without an r-fork. Order 24(3), 181–191 (2007). MR 2358080 (2008m:05294)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    De Bonis, A., Katona, G.O.H., Swanepoel, K.J.: Largest family without \(A\cup B\subseteq C\cap D\). J. Combin. Theory Ser. A 111(2), 331–336 (2005). MR 2156217 (2006e:05169)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Engel, K.: Sperner Theory, Encyclopedia of Mathematics. Cambridge University Press, Cambridge (1997)CrossRefGoogle Scholar
  8. 8.
    Erdös, P.: On a lemma of littlewood and offord. Bull. Amer. Math. Soc. 51, 898–902 (1945). MR 0014608 (7,309j)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goldman, J., Rota, G.-C.: The number of subspaces of a vector space. In: Tutte, W. (ed.) Recent Progress in Combinatorics, pp 75–83. Academic Press, New York and London (1969). Reprinted in [17, pages 217–225]Google Scholar
  10. 10.
    Goldman, J., Rota, G.-C.: On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions. Stud. Appl. Math. XLIX(3), 239–258 (1970). Reprinted in [17, pages 226–245]MathSciNetGoogle Scholar
  11. 11.
    Griggs, J.R., Katona, G.O.H.: No four subsets forming an N. J. Combin. Theory Ser. A 115(4), 677–685 (2008). MR 2407920 (2009b:05257)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Griggs, J.R., Li, W.-T., Lu, L.: Diamond-free families. J. Combin. Theory Ser. A 119(2), 310–322 (2012). MR 2860596MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Griggs, J.R., Lu, L.: On families of subsets with a forbidden subposet. Combin. Probab. Comput. 18(5), 731–748 (2009). MR 2534266 (2010k:05329)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Katona, G.O.H.: Forbidden intersection patterns in the families of subsets (introducing a method). Horizons of combinatorics. Bolyai Soc. Math. Stud., vol. 17, pp. 119–140. Springer, Berlin (2008). MR 2432530 (2010b:05181)Google Scholar
  15. 15.
    Katona, G.O.H., Tarján, T.G.: Extremal problems with excluded subgraphs in the n-cube. Graph theory (Łagów, 1981). Lecture Notes in Math., vol. 1018, pp. 84–93. Springer, Berlin (1983). MR 730637 (85h:05058)Google Scholar
  16. 16.
    Kramer, L., Martin, R.R., Young, M.: On diamond-free subposets of the Boolean lattice. J. Combin. Theory Ser. A 120(3), 545–560 (2013). MR 3007136MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kung, J.P.S. (ed.): Gian-Carlo Rota on Combinatorics, Contemporary Mathematicians. Birkhäuser Boston Inc., Boston (1995)Google Scholar
  18. 18.
    Manske, J., Shen, J.: Three layer \({Q}_{2}\)-free families in the boolean lattice. Order 30, 589–592 (2013)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Sperner, E.: Ein satz über Untermengen einer endlichen Mengen. Math. Z. 27, 544–548 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Thanh, H.T.: An extremal problem with excluded subposet in the Boolean lattice. Order 15(1), 51–57 (1998). MR 1652861 (2000a:06022)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Department of MathematicsPomona CollegeClaremontUSA

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