Sperner Families Research Articles

The linear chromatic number of a Sperner family

Let S be a finite set and S a complete Sperner family on S, i.e. a Sperner family such that every x∈S is contained in some member of S. The linear chromatic number of S, defined by Cıvan, is the smallest integer n with the property that there exists a function f:S→{1,…,n} such that if f(x)=f(y), then every set in S which contains x also contains y or every set in S which contains y also contains x. We give an explicit formula for the number of complete Sperner families on S of linear chromatic number 2. We also prove tight bounds on the number of elements in a Sperner family of given chromatic number, and prove that complete Sperner families of maximum linear chromatic number are far more numerous those of lesser linear chromatic number.

Profile polytopes of some classes of families

The profile vector of a family F of subsets of an n-element set is (f0,f1,…,fn) where fi denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. Erdős regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.

AZ-Identities and Strict 2-Part Sperner Properties of Product Posets

One of the central issues in extremal set theory is Sperner's theorem and its generalizations. Among such generalizations is the best-known BLYM inequality and the Ahlswede--Zhang (AZ) identity which surprisingly generalizes the BLYM inequality into an identity. Sperner's theorem and the BLYM inequality has been also generalized to a wide class of posets. Another direction in this research was the study of more part Sperner systems. In this paper we derive AZ type identities for regular posets. We also characterize all maximum 2-part Sperner systems for a wide class of product posets.

Tilted Sperner families

Let A be a family of subsets of an n-set such that A does not contain distinct sets A and B with |A∖B|=2|B∖A|. How large can A be? Our aim in this note is to determine the maximum size of such an A. This answers a question of Kalai. We also give some related results and conjectures.

On two-part Sperner systems for regular posets (Extended Abstract)

Abstract In the Boolean lattice the BLYM inequality holds with equality if and only if the Sperner family consists of one complete level of subsets. In this paper we extend this strict BLYM-property to a subclass of normal posets. On the basis of this result we prove a strict two-part Sperner theorem of the direct product of any two posets from the same subclass.

On Universal Cycles for new Classes of Combinatorial Structures

A universal cycle (u-cycle) is a compact listing of a collection of combinatorial objects. In this paper, we use natural encodings of these objects to show the existence of u-cycles for collections of subsets, restricted multisets, and lattice paths. For subsets, we show that a u-cycle exists for the k-subsets of an n-set if we let k vary in a non zero length interval. We use this result to construct a “covering” of length $(1+o(1))$$n \choose k$ for all subsets of $[n]$ of size exactly k with a specific formula for the $o(1)$ term. We also show that u-cycles exist for all n-length words over some alphabet $\Sigma,$ which contain all characters from $R \subset \Sigma.$ Using this result we provide u-cycles for encodings of Sperner families of size 2 and proper chains of subsets.

A tour of M-part L-Sperner families

In this paper we investigate common generalizations of more-part and L-Sperner families. We prove a BLYM inequality for M-part L-Sperner families and obtain results regarding the homogeneity of such families of maximum size through the convex hull method. We characterize those M-part Sperner problems, where the maximum family size is the classical ( n ⌊ n / 2 ⌋ ) . We make a conjecture on the maximum size of M-part Sperner families for the case of equal parts of size 2 ℓ − 1 and prove the conjecture in some special cases. We introduce the notion of k-fold M-part Sperner families, which specializes to the concept of M-part Sperner families for k = 1 , and generalize some M-part Sperner results to k-fold M-part Sperner families. We also approach the M-part Sperner problem from the viewpoints of graph product and linear programming, and prove the 2-part Sperner theorem using linear programming. This paper can be used as a survey, as in addition to the new results, problems and conjectures, we provide a number of alternative proofs, discuss at length a number of generalizations of Sperner's theorem, and for the sake of completeness, we give proofs to many simple facts that we use.

Open problems on linear sperner families and lattices

We introduce linear Sperner families and lattices, and provide some open problems.

Admissible values of one parameter for maximal Sperner families of subsets of the type (k, k + 1)

In this paper we generalize one assertion (obtained by us earlier) on admissible values of a certain parameter for partial maximal Sperner families (m. S. f.) of subsets of a finite set of the type (k, k + 1). We also prove that the minimal value of the parameter under consideration for all m. S. f. of the type (k, k + 1), except for two families, is less than \( \left( {_k^{n - 1} } \right) \) − 1.

A unified interpretation of several combinatorial dualities

Several combinatorial structures exhibit a duality relation that yields interesting theorems, and, sometimes, useful explanations or interpretations of results that do not concern duality explicitly. We present a common characterization of the duality relations associated with matroids, clutters (Sperner families), oriented matroids, and weakly oriented matroids. The same conditions characterize the orthogonality relation on certain families of vector spaces. This leads to a notion of abstract duality.

Weighted cross-intersecting families

In this paper we investigate weighted cross-intersecting families: if α , β > 0 are given constants, we want to find the maximum of α | A | + β | B | for A , B uniform cross-intersecting families. We determine the maximum sum, even if we have restrictions of the size of A . As corollaries, we will obtain some new bounds on the shadows and the shades of uniform families. We give direct proofs for these bounds, as well, and show that the theorems for cross-intersecting families also follow from these results. Finally, we will generalize the LYM inequality not only for cross-intersecting families, but also for arbitrary Sperner families.

An AZ-style identity and Bollobás deficiency

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.

All Maximum Size Two-Part Sperner Systems: In Short

In this note we give a very short proof for the description of all maximum size two-part Sperner systems.

EKR type inequalities for 4-wise intersecting families

Let 1 ⩽ t ⩽ 7 be an integer and let F be a k-uniform hypergraph on n vertices. Suppose that | A ∩ B ∩ C ∩ D | ⩾ t holds for all A , B , C , D ∈ F . Then we have | F | ⩽ ( n − t k − t ) if | k n − 1 2 | < ε holds for some ε > 0 and all n > n 0 ( ε ) . We apply this result to get EKR type inequalities for “intersecting and union families” and “intersecting Sperner families.”

A classification of the structures of some Sperner families and superimposed codes

In this paper we will give a new proof by using group action to prove the uniqueness of maximal Sperner families F max n of [ n ] . We will also prove the uniqueness of Sperner families F of [ n ] with | F | = n ⌊ n 2 ⌋ - 1 by using a combinatorial approach. Furthermore, by using the uniqueness of Sperner family, we will classify all the structures of ( 1 , 2 ) superimposed codes of size 9 × 10 and 9 × 11 .

Two-Part and k-Sperner Families: New Proofs Using Permutations

This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdos and G. O. H. Katona, J. Combin. Theory. Ser. A, 43 (1986), pp. 58--69; S. Shahriari, Discrete Math., 162 (1996), pp. 229--238] determining all extremal two-part Sperner families and for the uniqueness of k-Sperner families of maximum size [P. Erdos, Bull. Amer. Math. Soc., 51 (1945), pp. 898--902].

Quadratic LYM-type inequalities for intersecting Sperner families

Let $\mathcal{F}\subseteq 2^{[n]}$ be a intersecting Sperner family (i.e. $A \not\subset B, A \cap B \neq \emptyset$ for all $A,B \in \mathcal{F}$) with profile vector $(f_i)_{i=0 \ldots n}$ (i.e. $f_i=|\mathcal{F} \cap \binom{[n]}{i}|$). We present quadratic inequalities in the $f_i$'s which sharpen the previously known linear $\mathrm{LYM}$-type inequalities.

Polynomial Lym Inequalities

For a Sperner family A ⊂ 2[n] let A i denote the family of all i-element sets in A. We sharpen the LYM inequality $$ {\sum\nolimits_i {{\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} \leqslant 1} } $$ by adding to the LHS all possible products of fractions $$ {\left| {{\user1{\mathcal{A}}}_{i} } \right|}/{\left( {{}^{n}_{i} } \right)} $$ , with suitable coefficients. A corresponding inequality is established also for the linear lattice and the lattice of subsets of a multiset (with all elements having the same multiplicity).

Random walks and multiply intersecting families

Let F ⊂ 2 [ n ] be a 3-wise 2-intersecting Sperner family. It is proved that | F | ⩽ n - 2 ( n - 2 ) / 2 if n even , n - 2 ( n - 1 ) / 2 + 2 if n odd holds for n ⩾ n 0 . The unique extremal configuration is determined as well.

Extremal Theorems for Databases

We say, that a subset K of the columns of a matrix is called a key, if every two rows that agree in the columns of K agree also in all columns of the matrix. A matrix represents a Sperner system K, if the system of minimal keys is exactly K. It is known, that such a representation always exists. In this paper we show, that the maximum of the minimum number of rows, that are needed to represent a Sperner system of only two element sets is 3(n/3+o(n)). We consider this problem for other classes of Sperner systems (e.g., for the class of trees, i.e. each minimal key has cardinality two, and the keys form a tree), too. The concept of keys plays an important role in database theory.