An existence theorem for antichains

Let k₁, k₂,…, k_n be given integers, 1 ? k₁ ? k₂ ? … ? k_n, and let S be the set of vectors x = (x₁,…, x_n) with integral coefficients satisfying 0 ? x_i ? k_i, i = 1, 2, 3,…, n. A subset H of S is an antichain (or Sperner family or clutter) if and only if for each pair of distinct vectors x and y in H the inequalities x_i ? y_i, i = 1, 2,…, n, do not all hold. Let |H| denote the number of vectors in H, let K = k₁ + k₂ + … + k_n and for 0 ? l ? K let (l)H denote the subset of H consisting of vectors h = (h₁, h₂,…, h_n) which satisfy h₁ + h₂ + … + h_n = l. In this paper we show that if H is an antichain in S, then there exists an antichain H′ in S for which |(l)H′| = 0 if $\text{l <}\text{K}\text{2}$, $\text{|(}\text{K}\text{2}\text{)H′| = |(}\text{K}\text{2}\text{)H|}$ if K is even and |(l)H′| = |(l)H| + |(K ? l)H| if $\text{l>}\text{K}\text{2}$.     

On Erdős-Rado numbers

In this paper new proofs of the Canonical Ramsey Theorem, which originally has been proved by Erd?s and Rado, are given. These yield improvements over the known bounds for the arising Erd?s-Rado numbersER(k; l), where the numbersER(k; l) are defined as the least positive integern such that for every partition of thek-element subsets of a totally orderedn-element setX into an arbitrary number of classes there exists anl-element subsetY ofX, such that the set ofk-element subsets ofY is partitioned canonically (in the sense of Erd?s and Rado). In particular, it is shown that $$2^{c1} .l^2 \leqslant ER(2;l) \leqslant 2^{c_2 .l^2 .\log l} $$ for every positive integerl≥3, wherec ₁,c ₂ are positive constants. Moreover, new bounds, lower and upper, for the numbersER(k; l) for arbitrary positive integersk, l are given.     

On a problem of Katona on minimal separating systems

Let S be an n-element set. In this paper, we determine the smallest number f(n) for which there exists a family of subsets of S{A₁,A₂,…,A_f(n)} with the following property: Given any two elements x, y ∈ S (x ≠ y), there exist k, l such that A_k ∩ A_l= ?, and x ∈ A_k, y ∈ A_l. In particular it is shown that f(n)= 3 log₃n when n is a power of 3.     

More on the generalized macaulay theorem — II

Let k₁ ? k₂? ? ? k_n be given positive integers and let S denote the set of vectors x = (x₁, x₂, … ,x_n) with integer components satisfying 0 ? x₁ ? k_ni = 1, 2, …, n. Let X be a subset of S (l)X denotes the subset of X consisting of vectors with component sum l; F(m, X) denotes the lexicographically first m vectors of X; ?X denotes the set of vectors in S obtainable by subtracting 1 from a component of a vector in X; |X| is the number of vectors in X. In this paper it is shown that |?F(e, (l)S)| is an increasing function of l for fixed e and is a subadditive function of e for fixed l.     

A Fisher type inequality

In this paper we study subsets of a finite set that intersect each other in at most one element. Each subset intersects most of the other subsets in exactly one element. The following theorem is one of our main conclusions. Let S₁,… S_m be m subsets of an n-set S with |S₁| ? 2 (l = 1, …,m) and |S_i ∩ S_j| ? 1 (i ≠ j; i, j = 1, …, m). Suppose further that for some fixed positive integer c each S_i has non-empty intersection with at least m ? c of the remaining subsets. Then there is a least positive integer M(c) depending only on c such that either m ? n or m ? M(c).     

Intersection theorems for group divisible difference sets

Let D be an (m,n;k;λ₁,λ₂)-group divisible difference set (GDDS) of a group G, written additively, relative to H, i.e. D is a k-element subset of G, H is a normal subgroup of G of index m and order n and for every nonzero element g of G,?{(d₁,d₂)?,d₁,d₂?D,d₁?d₂=g}? is equal to λ₁ if g is in H, and equal to λ₂ if g is not in H. Let H₁,H₂,…,H_m be distinct cosets of H in G and S_i=D∩H_i for all i=1,2,…,m. Some properties of S₁,S₂,…,S_m are studied here. Table 1 shows all possible cardinalities of S_i's when the order of G is not greater than 50 and not a prime. A matrix characterization of cyclic GDDS's with λ₁=0 implies that there exists a cyclic affine plane of even order, say n, only if n is divisible by 4 and there exists a cyclic (n?1,$\text{1}\text{2}$n?1,$\text{1}\text{4}$n?1)-difference set.     

A Note on Uniquely lntersectable Graphs

The following conjecture of Alter and Wang is proven. Consider the intersection graph G_n,m,n?2m, determined by the family of all m-element subsets of an n-element set. Then any realization of G_n,m as an intersection graph by a family of sets satisfies |∪_iA_i|?n; and if |∪_iA_i|=n, then F must be the family of all m-element subsets of ∪_iA_i.     

More on the generalized Macaulay theorem

Let k₁ ? k₂ ? … ? k_n be given positive integers and let F denote the set of vectors (l₁, …, l_n) with integer components satisfying 0 ? l_i ? k_i, i = 1, 2, …, n. If H is a subset of F, let (l)H denote the subset of H consisting of those vectors with component sum l, and let C((l)H) denote the smallest [(l)H] elements of (l)F. The generalized Macaulay theorem due to the author and B. Lindström [3] shows that |Gamma;((C)(l)(H)|, ? |Γ(C((l)H))|, where Γ((l)H) is the setof vectors in F obtainable by subtracting l from a single component of a vector in (l)H. A method is given for computing [Γ(C((l)H)] in this paper. It is analogous to the method for computing |Γ(C(l)H))| in the k₁ = … = k_n = 1 case which has been given independently by Katona [4] and Kruskal [5].     

On the respective terms of the derived and the polycentral series of a free lie algabra and an ideal

Let F be a free Lie algebra of rank> 1 and S be an ideal of F. Denote by F_m and F_{n l},…,n_k the terms of the lower central and the polycentral series of F. The aim of this paper is to provide a sufficient condition for the quotient algebra F_{n l},…,n_k/S_{n l},…,n_k to be infinitely generated. The case F_m/S_m was studied in [6] for free groups and in [ 2 ] for free Lie algebras. In this paper the following main theorem is proved : If F = F2 = S, k > 1 and n_i > 1 for i=l,…, k, then F_{n l}…,n_k/S_{n l} is infinitely generated.     

General decomposition theorems form-convex sets in the plane

A setS inR ^dis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm?1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom?1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane. IfS is a closedm-convex set in the plane,m ≧ 2, the first question may be answered in one way by the following result: If there is some lineH supportingS at a pointp in the kernel ofS, thenS is a union ofm ? 1 convex sets. Using this result, it is possible to prove several decomposition theorems forS under varying conditions. Finally, an answer to the second question is given: Ifm≧3, thenS is a union of (m?1)³2 ^m?3 or fewer convex sets.     

The equivariant topology of stable Kneser graphs

The stable Kneser graph SG_n,k, n?1, k?0, introduced by Schrijver (1978) [19], is a vertex critical graph with chromatic number k+2, its vertices are certain subsets of a set of cardinality m=2n+k. Björner and de Longueville (2003) [5] have shown that its box complex is homotopy equivalent to a sphere, Hom(K₂,SG_n,k)?Sk. The dihedral group D2m acts canonically on SG_n,k, the group C₂ with 2 elements acts on K₂. We almost determine the (C₂×D2m)-homotopy type of Hom(K₂,SG_n,k) and use this to prove the following results.The graphs SG_2s,4 are homotopy test graphs, i.e. for every graph H and r?0 such that Hom(SG_2s,4,H) is (r−1)-connected, the chromatic number χ(H) is at least r+6.If k∉{0,1,2,4,8} and n?N(k) then SG_n,k is not a homotopy test graph, i.e. there are a graph G and an r?1 such that Hom(SG_n,k,G) is (r−1)-connected and χ(G)<r+k+2.     

The Ramsey number of paths with respect to wheels

For graphs G and H, the Ramsey numberR(G,H) is the smallest positive integer n such that every graph F of order n contains G or the complement of F contains H. For the path P_n and the wheel W_m, it is proved that R(P_n,W_m)=2n-1 if m is even, m?4, and n?(m/2)(m-2), and R(P_n,W_m)=3n-2 if m is odd, m?5, and n?(m-1/2)(m-3).     

Counting subsets of the random partition and the ‘Brownian Bridge’ process

Let Ω_m be the set of partitions, ω, of a finite m-element set; induce a uniform probability distribution on Ω_m, and define X_ms(ω) as the number of s-element subsets in ω. We alow the existence of an integer-valued function n=n^(m)(t), t?[0, 1], and centering constants b_ms, 0?s? m, such that

$\text{Z}{}^{\mathrm{(m)}}\text{(t)=}\text{∑}\text{s=0}\text{n(m)(t)}\text{(X}{}_{\mathrm{ms}}\text{?b}{}_{\mathrm{ms}}\text{)}\text{√}\text{∑}\text{s=0}\text{m}\text{b}{}_{\mathrm{ms}}$

converges to the ‘Brownian Bridge’ process in terms of its finite-dimensional distributions.     

Antichains in the set of subsets of a multiset

A set F of distinct subsets x of a finite multiset M (that is, a set with several different kinds of elements) is a c-antichain if for no c + 1 elements x₀, x₁, …, x_c of F does x₀ ? x₁ ? ··· ? x_c hold. The weight of F, wF, is the total number of elements of M in the various elements x of F. For given integers f and c, we find min wF, where the minimum is taken over all f-element c-antichains F. Daykin [9, 10] has solved this problem for ordinary sets and Clements [3] has solved it for multisets, but only for c = 1.     

Inequalities concerning numbers of subsets of a finite set

Let S(n) denote the set of subsets of an n-element set. For an element x of S(n), let Γx and Px denote, respectively, all (|x| ?1)-element subsets of x and all (|x| + 1)-element supersets of x in S(n). Several inequalities involving Γ and P are given. As an application, an algorithm for finding an x-element antichain $\text{X}{}^1$ in S(n) satisfying $\text{| YX}{}^1\text{| ? | YX |}$ for all x-element antichains X in S(n) is developed, where YX is the set of all elements of S(n) contained in an element of X. This extends a result of Kleitman [9] who solved the problem in case x is a binomial coefficient.     

The Littlewood-Richardson rule for wreath products with symmetric groups and the quiver of the category F≀FIn

We give a new proof for the Littlewood-Richardson rule for the wreath product F?S_n where F is a finite group. Our proof does not use symmetric functions but use more elementary representation theoretic tools. We also derive a branching rule for inducing the natural embedding of F?S_n to F?S_n+1. We then apply the generalized Littlewood-Richardson rule for computing the ordinary quiver of the category F?FI_n where FI_n is the category of all injective functions between subsets of an n-element set.     

Zeta function of the projective curveaY 21 =bX 21 +cZ 21 over a class of finite fields, for odd primesl

Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY ²¹ =bX ²¹ +cZ ²¹ defined over finite fields F_q such thatq = p ^α? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_q. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_q, as rational functions in the variablet, for distinct cases ofa, b, andc, in F _q ^* . Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves. Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_q. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).     

A generalization of the Kruskal-Katona theorem

Let k_n ? k_n?1 ? … ? k₁ be positive integers and let ($\text{i}\text{j}$) denote the coefficient of xⁱ in $\text{Π}\text{r=1}\text{j}\text{(1 + x + x}{}^2\text{+ … + x}{}^{\mathrm{kr}}\text{)}$. For given integers l, m, where 1 ? l ? k_n + k_n?1 + … + k₁ and $\text{1 ? m ? (}\text{n}\text{n}\text{)}$, it is shown that there exist unique integers m(l), m(l ? 1),…, m(t), satisfying certain conditions, for which $\text{m = (}\text{m(l)}\text{l}\text{+ (}\text{m(l?1)}\text{l?1}\text{) + … + (}\text{m(t)}\text{t}\text{)}$. Moreover, any m l-subsets of a multiset with k_i elements of type i, i = 1, 2,…, n, will contain at least $\text{(}\text{m(l)}\text{l?1}\text{) + (}\text{m(l?1)}\text{l?2}\text{) + … + (}\text{m(t)}\text{t?1}$ different (l ? 1)-subsets. This result has been anticipated by Greene and Kleitman, but the formulation there is not completely correct. If k₁ = 1, the numbers ($\text{j}\text{i}$) are binomial coefficients and the result is the Kruskal-Katona theorem.     

Intersecting sperner families and their convex hulls

Let ℱ be a family of subsets of a finite set ofn elements. The vector (f ₀, ...,f _n ) is called the profile of ℱ wheref _i denotes the number ofi-element subsets in ℱ. Take the set of profiles of all families ℱ satisfyingF ₁⊄F ₂ andF ₁∩F ₂≠0 for allF ₁,F ₂teℱ. It is proved that the extreme points of this set inR ⁿ⁺¹ have at most two non-zero components. Dedicated to Paul Erdős on his seventieth birthday