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LYM Inequalities for Graded Posets

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Lucas Sabalka, SLU

  • Combinatorics Seminar
When Thu, Feb 21, 2013
from 11:00 AM to 11:50 AM
Where RH 142
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 The LYM inequality for the boolean lattice says that if you give the subset S of {1,...,n} a weight of 1/(n choose ∣S∣) then the sum of the weights in an antichain is at most 1. This fact gives an easy proof of Sperner's Theorem, that the size of a largest antichain in the boolean lattice is (n choose n/2). In this talk we will discuss what happens when you use other weights on other graded posets. In particular, we will characterize weightings that give an LYM inequality or a strict LYM inequality, and (in some sense) determine when these can be used to find large antichains. If time permits, we will prove a more general version of a sharpening of the LYM inequality due to Aydinian and Peter Erdos, which is itself a generalization of a theorem of Ahlswede and Zhang. For the most part, these results follow easily by adapting proof techniques from previously know special cases.

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