A note on the Björner--Kalai theorem
Xiongfeng Zhan, Xueyi Huang
TL;DR
This work revisits the Björner–Kalai characterization of all possible pairs (f, β) of an f-vector and Betti sequence for simplicial complexes. It introduces an error function δ_k that measures the gap in the BK88 inequalities, and shows how δ_k can refine these inequalities to exact equalities via a main theorem relating ∂_k(n), ∂^k(n+m), and an admissible error ε_k. The authors define δ(f) and δ_+(f) to obtain sharper reformulations of the maximal and minimal constructions ψ(f) and φ(β), with equality characterized precisely when δ(f)=0. By combining these refinements with Kruskal–Katona and Björner’s 2011 results, they derive a new number-theoretic inequality on counts of odd square-free integers with a given number of prime factors, illustrating a deep link between combinatorial topology and analytic number theory.
Abstract
In 1988, Björner and Kalai used combinatorial shadow functions to characterize the maximal Betti sequence for a given $f$-vector and the minimal $f$-vector for a given Betti sequence. Their description of the maximal Betti sequence was expressed through a set of inequalities. In this paper, we introduce an error function $δ_k$ associated with the combinatorial shadow functions and use it to sharpen these inequalities into exact equalities. As a corollary, we obtain an equivalent form of Björner and Kalai's characterization of all possible pairs $(f,β)$ that can occur as the $f$-vector and Betti sequence of a simplicial complex. Moreover, combining our results with a previous result of Björner in 2011, we derive a new number-theoretic inequality concerning the count of odd square-free integers with a specified number of prime factors.