A Common Generalization to Theorems on Set Systems with <Emphasis Type="Italic">L</Emphasis>-intersections

In this paper, we provide a common generalization to the well-known Erdös–Ko–Rado Theorem, Frankl–Wilson Theorem, Alon–Babai–Suzuki Theorem, and Snevily Theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with k-wise L-intersections by Füredi and Sudakov [J. Combin. Theory, Ser. A, 105, 143–159 (2004)]. We will also derive similar results on L-intersecting families of subspaces of an n-dimensional vector space over a finite field F _q , where q is a prime power.     

The edge-diametric theorem in Hamming spaces

The maximum number of edges spanned by a subset of given diameter in a Hamming space with alphabet size at least three is determined. The binary case was solved earlier by Ahlswede and Khachatrian [A diametric theorem for edges, J. Combin. Theory Ser. A 92(1) (2000) 1–16].     

Size-Maximal Symmetric Difference-Free Families of Subsets of [n]

Union-free families of subsets of [n] = {1, . . . , n} have been studied in Frankl and Füredi (Eur J Combin 5:127–131, 1984). In this paper, we provide a complete characterization of maximal symmetric difference-free families of subsets of [n].     

An intersection theorem for four sets

Fix integers n,r?4 and let F denote a family of r-sets of an n-element set. Suppose that for every four distinct A,B,C,D∈F with |A∪B∪C∪D|?2r, we have A∩B∩C∩D≠∅. We prove that for n sufficiently large, , with equality only if _?F∈FF≠∅. This is closely related to a problem of Katona and a result of Frankl and Füredi [P. Frankl, Z. Füredi, A new generalization of the Erd?s-Ko-Rado theorem, Combinatorica 3 (3-4) (1983) 341-349], who proved a similar statement for three sets. It has been conjectured by the author [D. Mubayi, Erd?s-Ko-Rado for three sets, J. Combin. Theory Ser. A, 113 (3) (2006) 547-550] that the same result holds for d sets (instead of just four), where d?r, and for all n?dr/(d−1). This exact result is obtained by first proving a stability result, namely that if |F| is close to then F is close to satisfying _?F∈FF≠∅. The stability theorem is analogous to, and motivated by the fundamental result of Erd?s and Simonovits for graphs.     

The exact bound in the Erdös-Ko-Rado theorem

This paper contains a proof of the following result: ifn≧(t+1)(k?t?1), then any family ofk-subsets of ann-set with the property that any two of the subsets meet in at leastt points contains at most \(\left( {\begin{array}{*{20}c} {n - t} \\ {k - t} \\ \end{array} } \right)\) subsets. (By a theorem of P. Frankl, this was known whent≧15.) The bound (t+1)(k-t-1) represents the best possible strengthening of the original 1961 theorem of Erdös, Ko, and Rado which reaches the same conclusion under the hypothesisn≧t+(k?t) \(\left( {\begin{array}{*{20}c} k \\ t \\ \end{array} } \right)^3 \) . Our proof is linear algebraic in nature; it may be considered as an application of Delsarte’s linear programming bound, but somewhat lengthy calculations are required to reach the stated result. (A. Schrijver has previously noticed the relevance of these methods.) Our exposition is self-contained.     

Erd?s-Ko-Rado for three sets

Fix integers k?3 and n?3k/2. Let F be a family of k-sets of an n-element set so that whenever A,B,C∈F satisfy |A∪B∪C|?2k, we have A∩B∩C≠∅. We prove that with equality only when ?_F∈FF≠∅. This settles a conjecture of Frankl and Füredi [2], who proved the result for n?k²+3k.     

Representable orientations of the free spikes

All orientations of binary and ternary matroids are representable [R.G. Bland, M. Las Vergnas, Orientability of matroids, J. Combinatorial Theory Ser. B 24 (1) (1978) 94–123; J. Lee, M. Scobee, A characterization of the orientations of ternary matroids, J. Combin. Theory Ser. B 77 (2) (1999) 263–291]. In this paper we show that this is not the case for matroids that are representable over GF(p^k) where k2. Specifically, we show that there are orientations of the rank-k free spike that are not representable for all k4. The proof uses threshold functions to obtain an upper bound on the number of representable orientations of the free spikes.     

A note on a conjecture by Füredi

In [Z. Füredi, Turán type problems, in: Surveys in Combinatorics, Guildford, 1991, in: London Math. Soc. Lecture Note Ser., vol. 166, Cambridge Univ. Press, Cambridge, 1991, pp. 253-300, MR1161467 (93d:05072)], Füredi raised a conjecture about the maximum size of L-intersecting families. In this note, we address a variant of this conjecture. In particular, we show that for any Steiner triple system S on [k], there exist a family F of k-sets on [n] with |F|=Ω(n2+?) and such that for every F₀∈F the family is isomorphic to S.     

Extending Greene's theorem to directed graphs

The purpose of this note is to prove a theorem on acyclic digraphs, conjectured by Linial (J. Combin. Theory Ser. A, 30 (1981), 331–334), which includes Greene's theorem (J. Combin. Theory Ser. A, 20 (1976), 69–79) on decompositions of a partially ordered set into antichains.     

A filtration of -Catalan numbers

The operator of F. Bergeron, Garsia, Haiman and Tesler [F. Bergeron, A. Garsia, M. Haiman, G. Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods Appl. Anal. 6 (1999) 363–420] acting on the k-Schur functions [L. Lapointe, A. Lascoux, J. Morse, Tableaux atoms and a new Macdonald positivity conjecture, Duke Math. J. 116 (2003) 103–146; L. Lapointe, J. Morse, Schur functions analogs for a filtration of the symmetric functions space, J. Combin. Theory Ser. A 101 (2003) 191–224; L. Lapointe, J. Morse, Tableaux on k+1-cores, reduced words for affine permutations and k-Schur expansion, J. Combin. Theory Ser. A 112 (2005) 44–81] indexed by a single column has a coefficient in the expansion which is an analogue of the (q,t)-Catalan number with a level k. When k divides n we conjecture a representation theoretical model in this case such that the graded dimensions of the module are the coefficients of the (q,t)-Catalan polynomials of level k. When the parameter t is set to 1, the Catalan numbers of level k are shown to count the number of Dyck paths that lie below a certain Dyck path with q counting the area of the path.     

Inclusion matrices and chains

Given integers t, k, and v such that 0?t?k?v, let Wtk(v) be the inclusion matrix of t-subsets vs. k-subsets of a v-set. We modify slightly the concept of standard tableau to study the notion of rank of a finite set of positive integers which was introduced by Frankl. Utilizing this, a decomposition of the poset ²[v] into symmetric skipless chains is given. Based on this decomposition, we construct an inclusion matrix, denoted by , which is row-equivalent to Wtk(v). Its Smith normal form is determined. As applications, Wilson's diagonal form of Wtk(v) is obtained as well as a new proof of the well-known theorem on the necessary and sufficient conditions for existence of integral solutions of the system Wtkx=b due to Wilson. Finally we present another inclusion matrix with similar properties to those of which is in some way equivalent to Wtk(v).     

Linear Ramsey Numbers for Bounded-Degree Hypergrahps

We show that the Ramsey number is linear for every uniform hypergraph with bounded degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. [V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), pp. 239–243] showed in 1983. Our proof demonstrates the potential of a new regularity lemma by [Y. Ishigami, A simple regularization of hypergraphs, preprint, arXiv:math/0612838 (2006)].     

How many lines must miss d points in a linear space

It was remarked by P. Erd?s, J. C. Fowler, V. T. Sós, and R. M. Wilson, (J Combin Theory Ser A 38, 1985, 131–142), that a non‐degenerate finite linear space on v points has the following property. For every point P the number of lines not passing through P is at least with equality if and only if the linear space is a projective plane. We give a short proof for this result using an algebraic tool. The main purpose of the article however is to generalize this result in the direction that gives a lower bound for the number of lines not passing through a given number d of points. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 441–450, 2006     

Covering the alternating groups by products of cycle classes

Given integers k,l?2, where either l is odd or k is even, we denote by n=n(k,l) the largest integer such that each element of An is a product of k cycles of length l. For an odd l, k is the diameter of the undirected Cayley graph Cay(An,Cl), where Cl is the set of all l-cycles in An. We prove that if k?2 and l?9 is odd and divisible by 3, then . This extends earlier results by Bertram [E. Bertram, Even permutations as a product of two conjugate cycles, J. Combin. Theory 12 (1972) 368-380] and Bertram and Herzog [E. Bertram, M. Herzog, Powers of cycle-classes in symmetric groups, J. Combin. Theory Ser. A 94 (2001) 87-99].     

Properties of intersecting families of ordered sets

The first part of this paper deals with families of ordered k-tuples having a common element in different position. A quite general theorem is given and special cases are considered. The second part deals with t families of sets with some intersection properties, and generalizes results by Bollobás, Frankl, Lovász and Füredi to this case.     

On the weighted enumeration of alternating sign matrices and descending plane partitions

We prove a conjecture of Mills, Robbins and Rumsey [Alternating sign matrices and descending plane partitions, J. Combin. Theory Ser. A 34 (3) (1983) 340-359] that, for any n, k, m and p, the number of n×n alternating sign matrices (ASMs) for which the 1 of the first row is in column k+1 and there are exactly m −1?s and m+p inversions is equal to the number of descending plane partitions (DPPs) for which each part is at most n and there are exactly k parts equal to n, m special parts and p nonspecial parts. The proof involves expressing the associated generating functions for ASMs and DPPs with fixed n as determinants of n×n matrices, and using elementary transformations to show that these determinants are equal. The determinants themselves are obtained by standard methods: for ASMs this involves using the Izergin-Korepin formula for the partition function of the six-vertex model with domain-wall boundary conditions, together with a bijection between ASMs and configurations of this model, and for DPPs it involves using the Lindström-Gessel-Viennot theorem, together with a bijection between DPPs and certain sets of nonintersecting lattice paths.     

Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of \({\mathbb N}^{(r)}\) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For \(r=2\), Motzkin and Straus’ theorem confirms this conjecture. For \(r=3\), it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi’s conjecture holds for bigger ranges of m when \(r=3\). We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.     

r-Qsym is free over Sym

Our main result is a proof of the Florent Hivert conjecture [F. Hivert, Local action of the symmetric group and generalizations of quasi-symmetric functions, in preparation] that the algebras of r-Quasi-Symmetric polynomials in x₁,x₂,…,xn are free modules over the ring of Symmetric polynomials. The proof rests on a theorem that reduces a wide variety of freeness results to the establishment of a single dimension bound. We are thus able to derive the Etingof-Ginzburg [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 2 (2002) 555-566] Theorem on m-Quasi-Invariants and our r-Quasi-Symmetric result as special cases of a single general principle. Another byproduct of the present treatment is a remarkably simple new proof of the freeness theorem for 1-Quasi-Symmetric polynomials given in [A.M. Garsia, N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2) (2003) 217-263].