Abstract
We study a min-max relation conjectured by Saks and West: For any two posets P and Q the size of a maximum semiantichain and the size of a minimum unichain covering in the product P×Q are equal. As a positive result we state conditions on P and Q that imply the min-max relation. However, we also have an example showing that in general the min-max relation is false. This disproves the Saks-West conjecture.
The first and the fourth author were supported by Polish MNiSW grant N206 4923 38. The second and the third author were partially supported by DFG grant FE-340/7-2 and ESF EuroGIGA project GraDR.
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References
Dilworth, R.P.: A decomposition theorem for partially ordered sets. Ann. of Math. 51(2), 161–166 (1950)
Escamilla, E., Nicolae, A., Salerno, P., Shahriari, S., Tirrell, J.: On nested chain decompositions of normalized matching posets of rank 3. Order 28, 357–373 (2011)
Greene, C.: Some partitions associated with a partially ordered set. J. Combinatorial Theory Ser. A 20(1), 69–79 (1976)
Greene, C., Kleitman, D.J.: The structure of Sperner k-families. J. Combinatorial Theory Ser. A 20(1), 41–68 (1976)
Griggs, J.R.: Sufficient conditions for a symmetric chain order. SIAM J. Appl. Math. 32(4), 807–809 (1977)
Griggs, J.R.: On chains and Sperner k-families in ranked posets. J. Combin. Theory Ser. A 28(2), 156–168 (1980)
Griggs, J.R.: Matchings, cutsets, and chain partitions in graded posets. Discrete Math. 144(1-3), 33–46 (1995); Combinatorics of ordered sets (Oberwolfach, 1991)
Liu, Q., West, D.B.: Duality for semiantichains and unichain coverings in products of special posets. Order 25(4), 359–367 (2008)
Saks, M.: A short proof of the existence of k-saturated partitions of partially ordered sets. Adv. in Math. 33(3), 207–211 (1979)
Tovey, C.A., West, D.B.: Networks and chain coverings in partial orders and their products. Order 2(1), 49–60 (1985)
Trotter Jr., L.E., West, D.B.: Two easy duality theorems for product partial orders. Discrete Appl. Math. 16(3), 283–286 (1987)
Trotter, W.T.: Partially ordered sets. In: Handbook of Combinatorics, vol. 1, pp. 433–480. Elsevier, Amsterdam (1995)
Viennot, X.G.: Chain and antichain families, grids and Young tableaux. In: Orders: Description and Roles (L’Arbresle 1982). North-Holland Math. Stud., vol. 99, pp. 409–463. North-Holland, Amsterdam (1984)
Wang, Y.: Nested chain partitions of LYM posets. Discrete Appl. Math. 145(3), 493–497 (2005)
West, D.B.: Unichain coverings in partial orders with the nested saturation property. Discrete Math. 63(2-3), 297–303 (1987); Special issue: ordered sets (Oberwolfach, 1985)
West, D.B., Tovey, C.A.: Semiantichains and unichain coverings in direct products of partial orders. SIAM J. Algebraic Discrete Methods 2(3), 295–305 (1981)
Wu, C.: On relationships between semi-antichains and unichain coverings in discrete mathematics. Chinese Quart. J. Math. 13(2), 44–48 (1998)
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Bosek, B., Felsner, S., Knauer, K., Matecki, G. (2012). News about Semiantichains and Unichain Coverings. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds) Computer Science – Theory and Applications. CSR 2012. Lecture Notes in Computer Science, vol 7353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30642-6_5
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DOI: https://doi.org/10.1007/978-3-642-30642-6_5
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