Projection inequalities for antichains
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PROJECTION INEQUALITIES FOR ANTICHAINS
BY
Konrad Engel Institut f¨ ur Mathematik, Universit¨ at Rostock, 18051 Rostock, Germany e-mail: [email protected] AND
Themis Mitsis Department of Mathematics and Applied Mathematics, University of Crete 70013 Heraklion, Greece e-mail: [email protected] AND
Christos Pelekis∗ Institute of Mathematics, Czech Academy of Sciences ˇ Zitna 25, Praha 1, Czech Republic e-mail: [email protected] AND
Christian Reiher Fachbereich Mathematik, Universit¨ at Hamburg, Hamburg, Germany e-mail: [email protected]
∗ Research was supported by the Czech Science Foundation, grant number GJ16-
ˇ project 18-01472Y and RVO: 67985840. 07822Y, by GACR Received December 16, 2018 and in revised form June 17, 2019
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K. ENGEL ET AL.
Isr. J. Math.
ABSTRACT
Let n be an integer with n ≥ 2. A set A ⊆ Rn is called an antichain (resp. weak antichain) if it does not contain two distinct elements x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ) satisfying xi ≤ yi (resp. xi < yi ) for all i ∈ {1, . . . , n}. We show that the Hausdorff dimension of a weak antichain A in the n-dimensional unit cube [0, 1]n is at most n−1 and that the (n−1)-dimensional Hausdorff measure of A is at most n, which are the best possible bounds. This result is derived as a corollary of the following projection inequality, which may be of independent interest: The (n − 1)dimensional Hausdorff measure of a (weak) antichain A ⊆ [0, 1]n cannot exceed the sum of the (n − 1)-dimensional Hausdorff measures of the n orthogonal projections of A onto the facets of the unit n-cube containing the origin. For the proof of this result we establish a discrete variant of the projection inequality applicable to weak antichains in Zn and combine it with ideas from geometric measure theory.
1. Introduction Sperner’s theorem [22], a cornerstone of extremal set theory, determines for each positive integer n the maximal size of an antichain in the power set of an n-element set and describes the extremal configurations. In the statements that follow, [n] denotes the set {1, . . . , n} of the first n natural numbers and A ⊆ ℘([n]) is said to be an antichain if x ⊆ y holds for any distinct x, y ∈ A. Theorem 1.1 (Sperner): If n ≥ 1 is an integer and A ⊆ ℘([n]) is an antichain, n then |A| ≤ n/2 . Equality holds if and only if for some ∈ {n/2, n/2 } the set A is the collection of all -element subsets of [n]. Sperner’s fundamental result has been generalised in various ways and gave rise to a substantial body of future developments both within extremal set theory and beyond (see [1, 10]). Observe that via characteristic vectors the power set ℘([n]) can be identified with the set {0, 1}n of n-dimensional 0-1-vectors. Moreover, for any two subsets x and y of [n] we have x ⊆ y if and only if the characteristic vector of x is coordinate-wise at most the characteristic vector of y. Therefore, Sperner’s theorem can be reformulated as a statement about {0, 1}n equipped with the product partial ordering. It seems natural t
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