Order https://doi.org/10.1007/s11083-018-9480-5 A Simple Upper Bound on the Number of Antichains in [t ] n Shen-Fu Tsai 1 Received: 7 June 2017 / Accepted: 23 November 2018 / © Springer Nature B.V. 2018 Abstract In this paper for t> 2 and n> 2, we give a simple upper bound on a([t ] n ), the number of antichains in chain product poset [t ] n . When t = 2, the problem reduces to classi- cal Dedekind’s problem posed in 1897 and studied extensively afterwards. However few upper bounds have been proposed for t> 2 and n> 2. The new bound is derived with straightforward extension of bracketing decomposition used by Hansel for bound 3 ( n ⌊n/2⌋ ) for classical Dedekind’s problem. To our best knowledge, our new bound is the best when Θ  ( log 2 t ) 2  = 6t 4 (log 2 (t +1)) 2 π (t 2 −1)  2t − 1 2 log 2 (πt)  2 <n and t = ω  n 1/8 (log 2 n) 3/4  . Keywords Partially ordered set · Dedekind’s problem · Monotonic Boolean function 1 Introduction Definition 1 Let [t ]={1,...,t } and let [t ] n be the set of all tuples (x 1 ,...,x n ) such that x i ∈[t ] for each i . For tuples x = (x 1 ,...,x n ) ∈[t ] n and y = (y 1 ,...,y n ) ∈[t ] n , we say x ≤ y if x i ≤ y i for each i ∈[n]. A set of n-tuples forms an antichain if it does not contain distinct tuples x and y such that x ≤ y . 1.1 Dedekind’s Problem Dedekind [2] posed the question of estimating ψ (n), the number of antichains in Boolean lattice [2] n . He asked whether log 2 ψ (n) is asymptotic to its obvious lower bound ( n ⌊n/2⌋ ) . Later Hansel [4] obtained 3 ( n ⌊n/2⌋ ) as the upper bound on ψ (n) using bracketing decompo- sition of [2] n due to Greene and Kleitman [3] and the bijection between antichains, ideals, and monotone Boolean functions on [2] n . Kleitman and Markowsky [7] improved this upper bound to 2 ( n ⌊n/2⌋ )(1+O(log 2 n/n)) , followed by yet sharper estimates given by Korshunov [8]  Shen-Fu Tsai parity@gmail.com; parity@google.com 1 Google Inc., 747 6th Street South, Kirkland, WA, USA