The local dimension of suborders of the Boolean lattice

Abstract

We prove upper and lower bounds on the local dimension of any pair of layers of the Boolean lattice, and show that ldim Q n 1,⌊n/2⌋ ∼ n log 2 n as n → ∞. Previously, all that was known was a lower bound of Ω(n/ log n) and an upper bound of n.Improving a result of Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang, we also prove that that the maximum local dimension of an n-element poset is at least. We also show that there exist posets of arbitrarily large dimension whose dimension and local dimension are equal.

Local $t$-dimension

“…Kim et al [11] proved that, as n → ∞, there exist n-element posets with local dimension Ω n log n . Of course the same is true for local t-dimension for every t. The author [12] improved this lower bound by a constant factor, showing that there exists an n-element poset with local dimension (and hence local t-dimension for every t) at least n 4 log 3n for all n ≥ 2.…”

“…What is the exact value of ldim ⋆ t (n), for all integers n ≥ t ≥ 2? By an argument similar to the proof of Inequality 3 of Proposition 2 in [12], for any t ≥ 2 and all m, n ∈ N, ldim ⋆ t (mn) ≤ ldim ⋆ t (m) + ldim ⋆ t (n). It follows that, if ldim ⋆ t (m) < log t m for any m, then we can improve the trivial upper bound ldim ⋆ t (n) ≤ ⌈log t n⌉ by a constant factor for all n. However, we do not know of any examples of such m for any t.…”