Erdös Conjecture on Sums of Distinct Numbers

A conjecture of Erdös that a set of n distinct numbers having the most linear combinations with coefficients 0,1 all equal are n integers of smallest magnitude is here proven. The result follows from a theorem of Stanley that implies that the integers from n have the most such linear combinations having k distinct values for every k. The same result is shown to hold for complex numbers and vectors in Hilbert space. It is shown that the number of linear combinations taking on k distinct values is maximized by the same configuration, for every k. Generalization to the case in which irregular distinctness restrictions are imposed is also given.     

On the football pool problem for 6 matches: A new upper bound

Consider the set (F₃)⁶ of all 6 tuples x=(x₁,…, x₆) with x_i? {0, 1, 2}. We show that there exists a subset $\text{T}$ of (F₃)⁶ with 79 elements such that for any x of (F₃)⁶ there is an element if $\text{T}$ which differs from x in at most one coordinate.     

The Erdös–Hajnal Conjecture—A Survey

The Erdös–Hajnal conjecture states that for every graph H, there exists a constant such that every graph G with no induced subgraph isomorphic to H has either a clique or a stable set of size at least . This article is a survey of some of the known results on this conjecture.     

Extensions of the Erdös-Ko-Rado Theorem

An antichain of subsets of 1,2,...,n has the Erdös-Ko-Rado property if |A_i|?n/2 and A_i∩A_j≠Ø(i=j). This paper contains a number of results concerning the distribution of sizes of sets in such a family, and also in families where the restriction |A_i|?n/2 is removed.     

On a global upper bound for Jensen's inequality

In this paper we shall give a global upper bound for Jensen's inequality without restrictions on the target convex function f. We also introduce a characteristic c(f) i.e. an absolute constant depending only on f, by which the global bound is improved.     

The optimal upper bound of the number of generalized Euler configurations

Consider a generalized 3-body problem. The attraction force between any two bodies is proportional to the two masses and the b-th power of the mutual distance. Albouy and Fu have obtained the optimal upper bound of the number of generalized Euler configurations for the cases b 1 and b = 2, 3. This paper obtains the optimal upper bound for the remaining real values of b in a systematic way.     

On a generalization of an upper bound for the exponential function

With the introduction of a new parameter n, Kim recently generalized an upper bound for the exponential function that implies the inequality between the arithmetic and geometric means. In this paper, we answer some of Kim's conjectures about the inequalities between Kim's generalized upper bound and the original one. We also see the validity of Kim's generalization for some further negative values of x for the case in which the n is rational with both numerator and denominator odd. The range of its validity for negative x is investigated through the study of the zero distribution of a certain family of quadrinomials.     

On the upper bound of uniformly bounded solutions

New conditions of uniform boundedness of solutions are obtained and methods for calculating upper bounds are suggested.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 501–505, May, 1994.     

强色指数的一个新的上界

给出了图的强色指数的一个新的上界，并指出几类恰好达到该上界的图，从而改进了Erodoes和Nesetri的强色指数猜想，在某种意义上证明了这个猜想。     

A new upper bound on the queuenumber of hypercubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. In this paper, we show that the n-dimensional hypercube Q_n can be laid out using n−3 queues for n?8. Our result improves the previously known result for the case n?8.     

A new upper bound for the length of snakes

Ad-dimensional circuit code of spreads is a simple circuitC in the graph of thed-dimen sional unit cube with the property that for any verticesx andy ofC which differ in exactlyr co-ordinates,r<s, there exists a path fromx toy consisting ofr edges ofC. This property is useful for detecting and limiting errors. In this paper we give a new upper bound for the maximum length of ad-dimensional circuit code of spread 2.     

A new upper bound on the cyclic chromatic number

A cyclic coloring of a plane graph is a vertex coloring such that vertices incident with the same face have distinct colors. The minimum number of colors in a cyclic coloring of a graph is its cyclic chromatic number χ^c. Let Δ^* be the maximum face degree of a graph. There exist plane graphs with χ^c = ?3/2 Δ^*?. Ore and Plummer [ 5 ] proved that χ^c ≤ 2, Δ^*, which bound was improved to ?9/5, Δ^*? by Borodin, Sanders, and Zhao [ 1 ], and to ?5/3,Δ^*? by Sanders and Zhao [ 7 ]. We introduce a new parameter k^*, which is the maximum number of vertices that two faces of a graph can have in common, and prove that χ^c ≤ max {Δ^* + 3,k^* + 2, Δ^* + 14, 3, k^* + 6, 18}, and if Δ^* ≥ 4 and k^* ≥ 4, then χ^c ≤ Δ^* + 3,k^* + 2. © 2006 Wiley Periodicals, Inc. J Graph Theory     

A new upper bound for the complex Grothendieck constant

Let ? denote the real function $$\varphi (k) = k\smallint _0^{\pi /2} \frac{{cos^2 t}}{{\sqrt {1 - k^2 sin ^2 t} }}dt, - 1 \leqq k \leqq 1$$ and letK _G ^C be the complex Grothendieck constant. It is proved thatK _G ^C ≦8/π(k ₀+1), wherek ₀ is the (unique) solution to the equation?(k)=1/8π(k+1) in the interval [0,1]. One has 8/π(k ₀+1) ≈ 1.40491. The previously known upper bound isK _G ^C ≦e ^1?y ≈ 1.52621 obtained by Pisier in 1976.     

A new upper bound for the harmonious chromatic number

A harmonious coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colors in such a coloring. We obtain a new upper bound for the harmonious chromatic number of general graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 257–261, 1998     

A new upper bound for the bipartite Ramsey problem

We consider the following question: how large does n have to be to guarantee that in any two‐coloring of the edges of the complete graph K_n,n there is a monochromatic K_k,k? In the late 1970s, Irving showed that it was sufficient, for k large, that n ≥ 2^{k ? 1} (k ? 1) ? 1. Here we improve upon this bound, showing that it is sufficient to take where the log is taken to the base 2. © 2008 Wiley Periodicals, Inc. J Graph Theory 58: 351–356, 2008     

On an upper bound for the harmonious chromatic number of a graph

The upper bound for the harmonious chromatic number of a graph that has been given by Sin-Min Lee and John Mitchem is improved.