An AZ-style identity and Bollobás deficiency

Abstract

The powerful AZ identity is a sharpening of the famous LYM-inequality. More generally, Ahlswede and Zhang discovered a generalization in which the Bollobás inequality for two set families can be lifted to an identity.In this paper, we show another generalization of the AZ identity. The new identity implies an identity which characterizes the deficiency of the Bollobás inequality for an intersecting Sperner family. We also give some consequences relating to Helly families and LYM-style inequalities.

On Half-Way AZ-Style Identities

“…Proof of Theorem 4 The identity (7) is deduced immediately from (6) and Theorem 5 in [17]. Now we present a proof of (6).…”

“…We follow a similar process of the proof of Theorem 3 in [17]. Put W (X ) = (m + |X |) m + n m + |X | > 0 for all X ∈ U(A).…”

“…Moreover, the generalisation of the AZ identity [2] implies Bollobás inequality for two set systems [7]. Several results on AZ-style identities are also proposed; for example, AZ-style identities for other posets [3], pseudo-LYM inequalities and AZ identities [11], duals of AZ-identities [9,12], AZ type identity for k-chains in k-Sperner families [14], extremal cases of the Ahlswede-Cai identity [15], parametrised AZ identity [17].…”

“…In this paper, we present identities of "half-way extraction" from the recent generalization of AZ identities [17]. It means that summation in these identities is over subsets whose cardinalities are not exceedings n 2 .…”

Boundary optimization for rough sets

“…Assume that a * contains at most one item 6. Then by (5) 5m − n = 4i + (5 − b)j + (5 − (b + 1))k ≥ 4 + 0 − 1 > 0, a contradiction to n > 5m. Thus a * contains an item 1 and two items 6.…”