Path spectra for trees

The path spectrum of a graph is the set of lengths of all maximal paths in the graph. A set S of positive integers is spectral if it is the path spectrum of a tree. We characterize the spectral sets containing at most two odd integers (and arbitrarily many even ones) and obtain several necessary conditions for a set to be spectral. We show that for each even integer s≥2 at least 1/4 of all subsets of the set {2,3,…,s} are spectral and conjecture that all the subsets with at least 3s/4 integers are spectral.     

Distribution of averages of Ramanujan sums

The average value of a certain normalization of Ramanujan sums is determined in terms of Bernoulli numbers and odd values of the Riemann zeta function. The distribution of values and limiting behavior of such a normalization are then studied along subsets of Beurling type integers with positive density and sequences of moduli with constraints on the number of distinct prime factors.     

Sum-intersective sets

We study additive intersective properties of sparse subsets of integers. In particular, we prove that for every set S with counting function o(log n) there exists a set A with lower asymptotic density 1/2 such that A + A is disjoint from S. We also show that this result is best possible.     

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length 5 were constructed in Davis et al. (1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length 7. We construct a permutation of the integers avoiding arithmetic progressions of length 6. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length 4, sharpening the lower bound of $\frac{1}{3}$ from LeSaulnier and Vijay (2011).     

Patterns of quadratic residues and nonresidues for infinitely many primes

If S is a nonempty, finite subset of the positive integers, we address the question of when the elements of S consist of various mixtures of quadratic residues and nonresidues for infinitely many primes. We are concerned in particular with the problem of characterizing those subsets of integers that consist entirely of either (1) quadratic residues or (2) quadratic nonresidues for such a set of primes. We solve problem (1) and we show that problem (2) is equivalent to a purely combinatorial problem concerning families of subsets of a finite set. For sets S of (essentially) small cardinality, we solve problem (2). Related results and some associated enumerative combinatorics are also discussed.     

Chromatic Numbers of Some Distance Graphs

Mathematical Notes - For positive integers n > r > s, G(n, r, s) is the graph whose vertices are the r-element subsets of an n-element set, two subsets being adjacent if their...     

The enumeration of generalised alternating subsets with congruences

We show that a number of problems involving the enumeration of alternating subsets of integers may be solved by a direct method as special cases of the enumeration of configurations which we have termed (l, q)-sequences.     

Imbedding posets in the integers

An imbedding of a poset P in the integers is a one-to-one order presevring map from P into the integers. Such a map always exists when P is finite and, moreover, certain imbeddings of subsets of finite P can be extended to imbeddings of the whole of P. D. E. Daykin has asked when an imbedding in the integers of a finite subset of a countable poset can be extended to the whole poset. This paper answers Daykin's question and some related questions.     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

Quotient and product sets of thin subsets of the positive integers

We study the cardinalities of A/A and AA for thin subsets A of the set of the first n positive integers. In particular, we consider the typical size of these quantities for random sets A of zero density and compare them with the sizes of A/A and AA for subsets of the shifted primes and the set of sums of two integral squares.     

Divisibility problems for function fields

We investigate three combinatorial problems considered by Erd?s, Rivat, Sárközy and Schön regarding divisibility properties of sum sets and sets of shifted products of integers in the context of function fields. Our results in this function field setting are better than those previously obtained for subsets of the integers. These improvements depend on a version of the large sieve for sparse sets of moduli developed recently by the first and third-named authors.     

Almost Combinatorial Selector Sets

Classes of subsets of positive integers based on the notion of selector functions similar to the Jockusch selector of a semirecursive set are studied.     

Polynomial configurations in difference sets

We prove a quantitative result on the existence of linearly independent polynomial configurations in the difference set of sparse subsets of the integers. This result is achieved by first establishing a higher dimensional analogue of a theorem of Sárközy and then applying a simple lifting argument.     

分配格的分式扩张

首先借助整数扩充为有理数的办法构作一个分配格对它的滤子的分式扩张,然后用泛映射性质来刻画由任意分配格所构作的分式扩张,最后讨论了这种扩张的理想结构。     

An existence theory for loopy graph decompositions

Let v≥k≥1 and λ≥0 be integers. Recall that a (v, k, λ) block design is a collection ??of k‐subsets of a v‐set X in which every unordered pair of elements in X is contained in exactly λ of the subsets in ??. Now let G be a graph with no multiple edges. A (v, G, λ) graph design is a collection ??of subgraphs, each isomoprhic to G, of the complete graph K_v such that each edge of K_v appears in exactly λof the subgraphs in ??. A famous result of Wilson states that for a fixed simple graph G and integer λ, there exists a (v, G, λ) graph design for all sufficiently large integers v satisfying certain necessary conditions. Here, we extend this result to include the case of loops in G. As a consequence, we obtain the asymptotic existence of equireplicate graph designs. Applications of the equireplicate condition are given. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:280‐289, 2011     

图的分数κ-因子

给定图Ｇ＝（Ｖ,Ｅ）．设ａ和ｂ是两个非负整数．是一个函数．如果对所有的均成立,称 ｆ为 Ｇ的一个分数［ａ,ｂ］－ 因子． ａ＝ ｂ＝ κ时,称ｆ为 Ｇ的一个分数 ｋ＝因子．本文给出了一个图有分数 ｋ－因子的充分必要条件．     

Ergodic Averages for Independent Polynomials and Applications

Szemerédi's theorem states that a set of integers withpositive upper density contains arbitrarily long arithmeticprogressions. Bergelson and Leibman generalized this, showingthat sets of integers with positive upper density contain arbitrarilylong polynomial configurations; Szemerédi's theorem correspondsto the linear case of the polynomial theorem. We focus on thecase farthest from the linear case, that of rationally independentpolynomials. We derive results in ergodic theory and in combinatoricsfor rationally independent polynomials, showing that their behaviordiffers sharply from the general situation.     

Structural results for semigroup subsets defined by factorization properties dependent on \Omega functions

We study combinatorial properties of a class of subsets of semigroups with divisor theory motivated by the study of oscillations of counting functions of sets of algebraic integers with prescribed factorization properties.     

High density piecewise syndeticity of sumsets

Renling Jin proved that if A and B   are two subsets of the natural numbers with positive Banach density, then A+B$A+B$ is piecewise syndetic. In this paper, we prove that, under various assumptions on positive lower or upper densities of A and B  , there is a high density set of witnesses to the piecewise syndeticity of A+B$A+B$. Most of the results are shown to hold more generally for subsets of Z^d${\mathbb{Z}}^d$. The key technical tool is a Lebesgue density theorem for measure spaces induced by cuts in the nonstandard integers.