Boolean algebras and Lubell functions

Abstract

Let 2 [n] denote the power set of [n], where [n] = {1, 2, . . . , n}. A collection B ⊂ 2 [n] forms a d-dimensional Boolean algebra if there exist pairwise disjoint sets X 0 , X 1 , . . . , X d ⊆ [n], all non-empty with perhaps the exception of. Let b(n, d) be the maximum cardinality of a family F ⊂ 2 X that does not contain a d-dimensional Boolean algebra. Gunderson, Rödl, andIn this paper, we use the Lubell function as a new measurement for large families instead of cardinality. The Lubell value of a family of sets F with F ⊆ 2 [n] is defined by h n (F ) = F ∈F 1/ n |F | . We prove the following Turán type theorem. If F ⊆ 2 [n] contains no d-dimensional Boolean algebra, then h n (F ) ≤ 2(n + 1) 1−2 1−d for sufficiently large n. This result implies b(n, d) ≤ Cn −1/2 d · 2 n , where C is an absolute constant independent of n and d. With some modification, the ideas in Gunderson, Rödl, and Sidorenko's proof can be used to obtain this result. We apply the new bound on b(n, d) to improve several Ramsey-type bounds on Boolean algebras. We also prove a canonical Ramsey theorem for Boolean algebras.

Ramsey Numbers for Partially-Ordered Sets

“…For a poset P , let L n (P ) be the maximum value lu n (F ) among families F ⊆ B n such that F is P -free. The following result has been used implicitly by many authors such as Axenovich and Walzer [2, Theorem 6] and Johnston, Lu and Milans [26,Theorem 3]. Although the result is straightforward, we provide the explicit statement in the case of 1-uniform Boolean Ramsey numbers along with a proof for completeness.…”

Forbidden induced subposets of given height

“…In general, Gunderson, Rödl and Sidorenko [16] showed that b(n, d) = O d (2 n n −1/2 d ), where the constant hidden in O d (.) is super exponential in d. This was improved by Johnston, Lu and Milans [18] to b(n, d) = O(2 n n −1/2 d ). Here, we present a short proof of the latter bound b(n, d) = O(2 n n −1/2 d ), and show a similar bound for a natural generalization of the problem in grids.…”

Set families with forbidden subposets

“…Conjecture 1 is equivalent to the statement that for each poset P , the Turán function satisfies La * (n, P ) = O( n ⌊n/2⌋ ) = O(2 n / √ n). It is known (see [13]) that for each poset P , there exists α > 0 such that La * (n, P ) = O(2 n /n α ). In general, proving upper bounds on La * (n, P ) appears to be more challenging than proving upper bounds on La(n, P ).…”