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Projections of Antichains

Published version
Peer-reviewed

Repository DOI


Type

Article

Change log

Authors

Janzer, Barnabás 

Abstract

jats:pA subset A of Zn is called a weak antichain if it does not contain two elements x and y satisfying Misplaced &x_i&lt;y_ix_i&lt;y_i for all i. Engel, Mitsis, Pelekis and Reiher showed that for any weak antichain A, the sum of the sizes of its (n−1)-dimensional projections must be at least as large as its size |A|. They asked what the smallest possible value of the gap between these two quantities is in terms of  |A|. We answer this question by giving an explicit weak antichain attaining this minimum for each possible value of |A|. In particular, we show that sets of the form $$A_N={x\in\mathbb{Z}^n: 0\leq x_j\leq N-1 \textrm{ for all j and } x_i=0\textrm{ for some i}}$$ minimise the gap among weak antichains of size |AN|.  </jats:p>

Description

Keywords

4901 Applied Mathematics, 4904 Pure Mathematics, 49 Mathematical Sciences

Journal Title

The Electronic Journal of Combinatorics

Conference Name

Journal ISSN

1097-1440
1077-8926

Volume Title

27

Publisher

The Electronic Journal of Combinatorics
Sponsorship
Engineering and Physical Sciences Research Council (2261055)