Interpretation of σ-independence from the longest forbidden chain for r≥3

Ascertain a clear interpretation, for r ≥ 3, of why in the generalized LYM inequalities and corresponding bounds for families of r-decompositions with componentwise t_k-chain-free constraints, the parameter σ = (t_1 t_2 ··· t_r) / max{t_1, …, t_r} is independent of max{t_k}, so that the component with the longest forbidden chain does not affect the inequalities or the resulting upper bounds.

Background

The paper introduces generalized LYM inequalities for families of r-decompositions in several lattice settings where each component D_k is t_k-chain free, possibly with different t_k. The resulting bounds depend on σ = (t_1 t_2 ··* t_r) / max{t_1, …, t_r}, which is independent of the largest t_k.

The authors note that for r = 2 there is a simple explanation: if D_1 is t_1-chain free, then D_2 is automatically t_1-chain free, hence also t_2-chain free when t_1 ≤ t_2. For r ≥ 3, they state that a clear interpretation of why the longest forbidden chain is irrelevant to the bounds remains elusive.