Abstract
LetL be the set of subsets of a finite set ordered by inclusion. Sperner found the maximum size of an antichain in L. Later Kruskal proved a theorem stronger than Sperner’s result. It has many applications and yields representations of integers in terms of cascades. There are cascade inequalities, many of which can be proved using Daykin’s algorithm, but the proofs of others are based on manipulation of binomial coefficients.
I will describe some applications of the above and also describe how both Leeb and Clement have proved a version of Kruskal’s theorem for different generalizations of L. In each generalization we wish to solve the problems already solved in L, and this leads to an axiomatic approach.
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Daykin, D.E. (1985). Ordered Ranked Posets, Representations of Integers and Inequalities from Extremal Poset Problems. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_10
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DOI: https://doi.org/10.1007/978-94-009-5315-4_10
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