Local dimension of a Boolean lattice

For every integer $n$ with $n \geq 4$, we prove that the local dimension of a poset consisting of all the subsets of ${1,\dots,n}$ equipped with the inclusion relation is strictly less than $n$, answering a question of Kim, Martin, Masařík, Shull, Smith, Uzzell, and Wang (Eur. J. Comb. 2020). We also study several related problems.

💡 Research Summary

The paper investigates the local dimension (ldim) of Boolean lattices and related posets, providing new upper bounds and exact asymptotics. The authors first recall that for any poset P, the local dimension is at most its classical dimension, and that the Boolean lattice Bₙ (the set of all subsets of {1,…,n} ordered by inclusion) satisfies dim(Bₙ)=n. The central result is that for every integer n ≥ 4, the local dimension of Bₙ is strictly smaller than n. To prove this, they construct explicit families of partial linear extensions (PLEs) for the small cases B₄ and B₇, showing ldim(B₄)≤3 and ldim(B₇)≤5. Using the subadditivity property ldim(P×Q) ≤ ldim(P)+ldim(Q) (Lemma 6) and the product decomposition Bₙ = (B₄)ᵏ×Bᵣ (where n = 4k+r, 0≤r<4), they obtain the general bound ldim(Bₙ) ≤ ⌈3n/4⌉ < n. The constructions were verified with a SAT solver and a Python script, and the code is publicly released.

The paper then turns to a restricted subposet B₁ₙ consisting of all non‑empty subsets together with the rule that a singleton {x} is comparable only to supersets that do not contain x. Prior work gave a lower bound Ω(n log n) for ldim(B₁ₙ). The authors present a matching upper bound, proving ldim(B₁ₙ) ≤ 2n log n + 3, thereby establishing ldim(B₁ₙ)=Θ(n log n). Their construction partitions the ground set into blocks of size d≈log n−log log n, and for each block creates 2ᵈ PLEs that order all larger sets below the appropriate singletons. Careful counting shows that each element appears in at most max{2d+1, r+2} PLEs, which yields the claimed bound.

Next, the authors explore the relationship between the size of a local realizer (the number of PLEs) and its frequency (the maximum number of times any element appears). They conjecture that an optimal local realizer can be chosen with size bounded by a function of the classical dimension dim(P). However, Theorem 4 demonstrates that for B₁ₙ any local realizer of size at most c n must have frequency at least n·2(c+1), implying that any such bounding function must be super‑linear. This result uses Turán’s theorem to relate the edge density of a graph derived from the realizer to the existence of large independent sets, which in turn force high frequency.

Finally, the paper extends the analysis to multiset lattices Mₙ, where elements are multisets over