A Class of lattices and boolean functions related to a Manickam-Mikl'os-Singhi Conjecture

Let n, r be two fixed integers such that 0 ≤ r ≤ n and let I n = {1, 2 • • • , n}. In the first part of this paper (Section 1) we define a partial order ⊑ on the power set P(I n ) having the following property : if X, Y are two subsets of I n such that X ⊑ Y , then i∈X a i ≤ i∈Y a i , for each n-multiset {a 1 , • • • , a n } of real numbers such that a 1 ≥ • • • ≥ a r ≥ 0 > a r+1 ≥ • • • ≥ a n . This order defines a lattice structure on P(I n ) that we will denote by (S(n, r), ⊑). We show as this lattice is distributive, graded (Section 2), involutive (Section 3), i.e. X ⊑ Y implies Y c ⊑ X c , and we also give an algorithmic method to generate uniquely its Hasse diagram (Section 4) and a recursive formula to count the number of its elements having fixed rank (Section 5).

In the second part of the paper (Section 6) we establish the connection between the lattice S(n, r) and some combinatorial extremal sum problems related to a conjecture of Manickam, Miklös and Singhi. We give an interpretation of these problems in terms of a particular class of boolean maps defined on S(n, r) (Section 7). Now we briefly summarize the historical motivations that have led us to build the lattice S(n, r) and the other associated structures.

In [21] the authors asked the following question: let n be an integer strictly greater than 1 and a 1 , • • • a n be real numbers satisfying the property n 1=1 a i ≥ 0. We may ask: how many of the subsets of the set {a 1 , • • • , a n } will have a non-negative sum? Following the notations of [21], the authors denote with A(n) the minimum number of the non-negative partial sums of a sum n i=1 a i ≥ 0, not counting the empty sum, if we take all the possible choices of the a i ’s. They prove (see Theorem 1 in [21]) that A(n) = 2 n-1 and they explain as Erdös, Ko and Rado investigated a question with an answer similar to this one: what is the maximum number of pairwise intersecting subsets of a n-elements set? As in their case, here also the question becomes more difficult if we restrict ourselves to the d-subsets. More details about this remark can be find in the famous theorem of Erdös-Ko-Rado [15] (see also [16] for an easy proof of it). Formally, with the introduction of the positive integer d, the problem is the following. Let 1 ≤ d < n be an integer; a function f : I n → R is called a n-weight function if x∈In f (x) ≥ 0. Denote with W n (R) the set of all the n-weight functions and if f ∈ W n (R) we set

and furthermore

In [8], Bier and Manickam proved that ψ(n, d)

Both the proofs use the Baranyai theorem on the factorization of complete hypergraphs [4] (see also [24] for a modern exposition of the theorem). In [21] and [20] it was conjectured that ψ(n, d) ≥ n-1 d-1 if n ≥ 4d. In [20] this conjecture has been set in the more general context of the association schemes (see [3] for general references on the subject). In the sequel we will refer to this conjecture as the Manickam-Miklös-Singhi (MMS) Conjecture. This conjecture is connected with the first distribution invariant of the Johnson association scheme (see [8], [20], [18], [19]). The distribution invariants were introduced by Bier [7], and later investigated in [9], [17], [18], [20]. In [20] the authors claim that this conjecture is, in some sense, dual to the theorem of Erdös-Ko-Rado [15]. Moreover, as pointed out in [22], this conjecture settles some cases of another conjecture on multiplicative functions by Alladi, Erdös and Vaaler, [2]. Partial results related to the Manickam-Miklös-Singhi conjecture have been obtained also in [5], [6], [11], [12], [13]. Now, if 1 ≤ r ≤ n, we set: (1) γ(n, r) = min{α(f ) :

The numbers γ(n, d, r) have been introduced in [11] and they also have been studied in [12], in order to solve the Manickam-Miklös-Singhi conjecture, because it is obviuos that:

(3) ψ(n, d) = min{γ(n, d, r) : 1 ≤ r ≤ n}.

Therefore the complete computation of these numbers gives an answer to the MMS conjecture but this is not the purpose of this paper.

In [21] it has been proved that γ(n, r) ≥ 2 n-1 for each r, and that γ(n, 1) = 2 n-1 .

Question 0.1. Is it true that γ(n, r) = 2 n-1 for each r?

Let us observe now that when we have a n-weight function f , the standard ways to produce n-subsets on which f takes non-negative values are the following :

(i) if we know that X and Y are two subsets of I n such that x∈X f (x) ≥ 0 and

Then we ask: A) Is it possible to axiomatize the properties (i) and (ii) in some type of abstract structure in such a way that the sum extremal problems upon described become particular extremal problems of more general problems?

B) In such abstract structure can we find unexpected links with other theories which help us to solve these sum extremal problems? C) Is it possible to define an algorithmic strategy in such abstract structure to approach these sum extremal problems in a deterministic way?

In this paper we show that the answer to all the previous questions is affirmative. We define

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