Abstract
This paper is concerned with some combinatorial problems related to the following theorem on partially ordered sets:
The minimal number of chains in the representation of a finite partially ordered set P as a set union of chains is equal to the maximal number of mutually non-comparable elements of P.
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References
R.P. Dilworth, A decomposition theorem for partially ordered sets, Ann. of Math. vol. 51 (1950) pp. 161–166.
G.B. Dantzig and A. Hoffman, On a theorem of Dilworth, Contributions to linear inequalities and related topics, Annals of Mathematics Studies, no. 38.
D.R. Fulkerson, Note on Dilworth’s decomposition for partially ordered sets, Proc. Amer. Math. Soc. vol. 7 (1956) pp. 701–702.
P. Hall, On representatives of subsets, J. London Math. Soc. vol. 10 (1935) pp. 26–30.
D. König, Theorie der Graphen, New York, Chelsea, 1950.
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Dilworth, R.P. (1990). Some Combinatorial Problems on Partially Ordered Sets. In: Bogart, K.P., Freese, R., Kung, J.P.S. (eds) The Dilworth Theorems. Contemporary Mathematicians. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4899-3558-8_2
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DOI: https://doi.org/10.1007/978-1-4899-3558-8_2
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