Abstract
We prove a “supersaturation-type” extension of both Sperner’s Theorem (1928) and its
generalization by Erd˝os (1945) to k-chains. Our result implies that a largest family whose
size is x more than the size of a largest k-chain free family and that contains the minimum
number of k-chains is the family formed by taking the middle (k ¡1) rows of the Boolean
lattice and x elements fromthe kth middle row. We prove our result using the symmetric chain
decomposition method of de Bruijn, van Ebbenhorst Tengbergen, and Kruyswijk (1951).
Original language | English |
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Publisher | Centre pour la Communication Scientifique Directe (CNRS), hal-00802000 |
Number of pages | 6 |
Publication status | Published - 2013 |