Representing posets with k-tournaments

In this paper, we introduce a binary relation on the vertex set of a k-tournament, and using this relation show that every finite poset with cardinality n4 can be represented by a k-tournament for every k satisfying 3kn–1.     

Maximum antichains in the product of chains

Let P be the poset k ₁ × ... × k _n , which is a product of chains, where n1 and k ₁ ... k _n 2. Let . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M1. This is a generalization of a classical result, Sperner's Theorem, which is the case k ₁= ... =k _n =2. We also determine the number and location of the maximum ranks of P.Research supported in part by the National Science Foundation 10/25/83.     

The Colin de Verdière number and sphere representations of a graph

Colin de Vedière introduced an interesting linear algebraic invariant (G) of graphs. He proved that (G)2 if and only ifG is outerplanar, and (G)3 if and only ifG is planar. We prove that if the complement of a graphG onn nodes is outerplanar, then (G)n–4, and if it is planar, then (G)n–5. We give a full characterization of maximal planar graphs whose complementsG have (G)=n–5. In the opposite direction we show that ifG does not have twin nodes, then (G)n–3 implies that the complement ofG is outerplanar, and (G)n–4 implies that the complement ofG is planar.Our main tools are a geometric formulation of the invariant, and constructing representations of graphs by spheres, related to the classical result of Koebe about representing planar graphs by touching disks. In particular we show that such sphere representations characterize outerplanar and planar graphs.     

Room designs and one-factorizations

The existence of a Room square of order 2n is known to be equivalent to the existence of two orthogonal one-factorizations of the complete graph on 2n vertices, where orthogonal means any two one-factors involved have at most one edge in common. DefineR(n) to be the maximal number of pairwise orthogonal one-factorizations of the complete graph onn vertices.The main results of this paper are bounds on the functionR. If there is a strong starter of order 2n–1 thenR(2n) 3. If 4n–1 is a prime power, it is shown thatR(4n) 2n–1. Also, the recursive construction for Room squares, to obtain, a Room design of sidev(u – w) +w from a Room design of sidev and a Room design of sideu with a subdesign of sidew, is generalized to sets ofk pairwise orthogonal factorizations. It is further shown thatR(2n) 2n–3.     

Faulkner geometry

Axioms are presented for a Barbilian geometry of dimension n2 over a ring for which ab=1 implies ba=1. It is shown that any Faulkner geometry of dimension n3 is coordinatized by a unique associative two-sided units ring R and that the group generated by all transvections is a group of Steinberg type over R.     

Sumsets Containing Powers of an Integer

Let n1, and let m be an integer with m2. We show that if a subset A of the interval [0,n] satisfies that 0A and |A|>1+n/2, then mA, the set of the sum of m (not necessarily distinct) elements in A, has a power of m. This result is best possible in the case that m is odd.     

A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for Cartesian powers P ⁿ of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L ⁿ . Namely, we show that under certain conditions the shadow of any initial segment of the order L ⁿ for n 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

Markov approximation of homogeneous lattice random fields

We refine some well-known approximation theorems in the theory of homogeneous lattice random fields. In particular, we prove that every translation invariant Borel probability measure on the space X of finite-alphabet configurations on ^d, d1, can be weakly approximated by Markov measures _n with supp(_n)=X and with the entropies h(_n)h(). The proof is based on some facts of Thermodynamic Formalism; we also present an elementary constructive proof of a weaker version of this theorem.Mathematics Subject Classifications (2000): Primary 28D20, 37C85, 60G60; secondary 82B20Dedicated to Professor A. I. Vorobyov, member of the Russian Academy of Sciences and Director of the Hematology Research Center of the Russian Academy of Medical Sciences, on the occasion of his 75th birthday     

The dimension theorem in axiomatic geometry

Given a matroid and an integer n 0, eleven conditions are shown to be equivalent to the validity of the rank formula r(E F) + r(E F = r(E) + r(F) for subspaces satisfying r(EF)n. For n=0 one finds the projective geometries. The case n=1 also includes the affine and the hyperbolic geometries, the case n=2 the Möbius geometries. The general case covers the incidence geometries of grade n of Wille.     

Random graphs and covering graphs of posets

For a graph G, we define c(G) to be the minimal number of edges we must delete in order to make G into a covering graph of some poset. We prove that, if p=n ^-1+(n) ,where (n) is bounded away from 0, then there is a constant k ₀>0 such that, for a.e. G _p , c(G _p )k ₀ n ¹⁺⁽ⁿ⁾ .In other words, to make G _p into a covering graph, we must almost surely delete a positive constant proportion of the edges. On the other hand, if p=n ^-1+(n) , where (n)0, thenc(G _p )=o(n ¹⁺⁽ⁿ⁾ ), almost surely.Partially supported by MCS Grant 8104854.     

Circular Chromatic Number and Mycielski Graphs

As a natural generalization of graph coloring, Vince introduced the star chromatic number of a graph G and denoted it by ^*(G). Later, Zhu called it circular chromatic number and denoted it by _c(G). Let (G) be the chromatic number of G. In this paper, it is shown that if the complement of G is non-hamiltonian, then _c(G)=(G). Denote by M(G) the Mycielski graph of G. Recursively define M^m(G)=M(M^m–1(G)). It was conjectured that if mn–2, then _c(M^m(K_n))=(M^m(K_n)). Suppose that G is a graph on n vertices. We prove that if , then _c(M(G))=(M(G)). Let S be the set of vertices of degree n–1 in G. It is proved that if |S| 3, then _c(M(G))=(M(G)), and if |S| 5, then _c(M²(G))=(M²(G)), which implies the known results of Chang, Huang, and Zhu that if n3, _c(M(K_n))=(M(K_n)), and if n5, then _c(M²(K_n))=(M²(K_n)).* Research supported by Grants from National Science Foundation of China and Chinese Academy of Sciences.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

Maximal Total Length Of k Disjoint Cycles In Bipartite Graphs

Let k, s and n be three integers with sk2, n2k+1. Let G=(V ₁,V ₂;E) be a bipartite graph with |V ₁|=|V ₂|=n. If the minimum degree of G is at least s+1, then G contains k vertex-disjoint cycles covering at least min(2n,4s) vertices of G.     

On mappings preserving equilateral triangles

Let E be a Euclidean space, dim E 2. We say that f : E E preserves equilateral triangles if for all triples of points x, y, z E we have We show that if E is a finite-dimensional Euclidean space, dim E 2, f:E E is measurable and preserves equilateral triangles, then it is a similarity transformation (an isometry multiplied by a positive constant). Moreover, in spaces at least three-dimensional we get a similarity transformation without any regularity assumption. Some generalizations as well as some interesting examples are also presented in the paper.     

The number of different distances determined by a set of points in the Euclidean plane

In 1946 P. Erdös posed the problem of determining the minimum numberd(n) of different distances determined by a set ofn points in the Euclidean plane. Erdös provedd(n) cn ^1/2 and conjectured thatd(n)cn/ logn. If true, this inequality is best possible as is shown by the lattice points in the plane. We showd(n)n ^4/5/(logn) ^c .The research of W. T. Trotter was supported in part by the National Science Foundation under DMS 8713994 and DMS 89-02481.     

Simple minimum coverings ofK n with copies ofK 4 −e

Summary AK ₄–e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofK _n (the complete undirected graph onn vertices) with vertex setS, into copies ofK ₄–e, the graph on four vertices with five edges. It is well-known [1] thatK ₄–e designs of ordern exist for alln 0 or 1 (mod 5),n 6, and that if (S, B) is aK ₄–e design of ordern then |B| =n(n – 1)/10.Asimple covering ofK _n with copies ofK ₄–e is a pair (S, C) whereS is the vertex set ofK _n andC is a collection of edge-disjoint copies ofK ₄–e which partitionE(K_n)P, for some . Asimple minimum covering ofK _n (SMCK _n) with copies ofK ₄–e is a simple covering whereP consists of as few edges as possible. The collection of edgesP is called thepadding. Thus aK ₄–e design of ordern isSMCK _n with empty padding.We show that forn 3 or 8 (mod 10),n 8, the padding ofSMCK _n consists of two edges and that forn 2, 4, 7 or 9 (mod 10),n 9, the padding consists of four edges. In each case, the padding may be any of the simple graphs with two or four edges respectively. The smaller cases need separate treatment:SMCK ₅ has four possible paddings of five edges each,SMCK ₄ has two possible paddings of four edges each andSMCK ₇ has eight possible paddings of four edges each.The recursive arguments depend on two essential ingredients. One is aK ₄–e design of ordern with ahole of sizek. This is a triple (S, H, B) whereB is an edge-disjoint collection of copies ofK ₄–e which partition the edge set ofK _n\K_k, whereS is the vertex set ofK _n, and is the vertex set ofK _k. The other essential is acommutative quasigroup with holes. Here we letX be a set of size 2n 6, and letX = {x ₁, x₂, ..., x_n} be a partition ofX into 2-element subsets, calledholes of size two. Then a commutative quasigroup with holesX is a commutative quasigroup (X, ) such that for each holex _i X, (x_i, ) is a subquasigroup. Such quasigroups exist for every even order 2n 6 [4].     

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Gröbner bases is applied to study triangulations of the second hypersimplex (2,n). We present a quadratic Gröbner basis for the associated toric idealK(K _n ). The simplices in the resulting triangulation of (2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK _n . Forn6 the number of distinct initial ideals ofI(K _n ) exceeds the number of regular triangulations of (2,n); more precisely, the secondary polytope of (2,n) equals the state polytope ofI(K _n ) forn5 but not forn6. We also construct a non-regular triangulation of (2,n) forn9. We determine an explicit universal Gröbner basis ofI(K _n ) forn8. Potential applications in combinatorial optimization and random generation of graphs are indicated.Research partially supported by a doctoral fellowship of the National University of Mexico, the National Science Foundation, the David and Lucile Packard Foundation and the U.S. Army Research Office (through ACSyAM/MSI, Cornell).     

On the stability of quasidouble step spline approximations for solutions ofy (n) = f(x,y, ⋯, y (n−1) )

In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C ^m–2 , to the initial value problemy ⁽ⁿ⁾ = f(x, y, , y ^(n–1) ). It will be shown that the method is unstable and hence divergent form n + 4.