Arie Bialostocki

On the Erdos-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

Abstract A link between Ramsey numbers for stars and matchings and the Erdos-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e ( K 2 m −1 )→{0, 1,…, m −1} is a mapping of the edges of the complete graph on 2 m −1 vertices into {0, 1,…, m −1}, then there exists a star K 1,m in K 2 m −1 with edges e 1 , e 2 ,…, e m such that c ( e 1 )+ c ( e 2 )+⋯+ c ( e m )≡0 (mod m ). Theorem 8. Let m be an integer. If c : e ( K r ( r +1) m −1 )→{0, 1,…, m −1} is a mapping of all the r-subsets of an ( r +1) m −1 element set S into {0, 1,…, m −1}, then there are m pairwise disjoint r-subsets Z 1 , Z 2 ,…, Z m of S such that c ( Z 1 )+ c ( Z 2 )+⋯+ c ( Z m )≡0 (mod m ).

Monochromatic and zero-sum sets of nondecreasing diameter

For positive integers m and r define f(m,r) to be the minimum integer n such that for every coloring of 1, 2, ?, n with r colors, there exist two monochromatic subsets B1, B2 ? 1, 2, ?, n (but not necessarily of the same color) which satisfy: (i) |B1¦=|B2|=m; (ii) The largest number in B1 is smaller than the smallest number in B2; (iii) The diameter of the convex hull spanned by B1 does not exceed the diameter of the convex hull spanned by B2. We prove that f(m, 2)=5m-3,f(m, 3)=9m-7 and 12m-9?f (m, 4) ?13m-11. Asymptotically, it is shown that e1mr?f(m,r)?c2mr log2r, where c1 and c2 are positive constants. Next we consider the corresponding questions for zero-sum sets and we generalize some of our results in the sense of the Erd?s?Ginzburg?Ziv theorem. Moreover, stronger versions are derived when the group under consideration is cyclic of prime order.

Note: Disjoint chorded cycles in graphs

We propose the following conjecture to generalize results of Posa and of Corradi and Hajnal. Let r,s be nonnegative integers and let G be a graph with |V(G)|>=3r+4s and minimal degree @d(G)>=2r+3s. Then G contains a collection of r+s vertex disjoint cycles, s of them with a chord. We prove the conjecture for r=0,s=2 and for s=1. The corresponding extremal problem, to find the minimum number of edges in a graph on n vertices ensuring the existence of two vertex disjoint chorded cycles, is also settled.

Some notes on the Erdos-Szekeres theorem

Abstract In the spirit of the Erdos-Szekeres theorem of 1935 we prove some canonical Ramsey Theorems in the plane. In Theorem 1 we are concerned with the number of points in the interior of a convex n -gon. In Theorem 2 we consider a set of points in general position along with a function from the set of points into the plane and prove the existence of a certain canonical configuration. A one-dimensional analog of Theorem 2 is a reformulation of a known theorem concerning intervals on the real line.

A zero-sum theorem

We determine the smallest integer n for which the following holds: if G is a nontrivial abelian group of order m, then every coloring of the integer set {1,2, ..., n} by the elements of G, results a zero-sum solution to x1 + x2 + ... + xm-1 < xm. It turns out that n depends only on the order of G and is equal to m(m - 1) + 1. If G is cyclic, then we get an Erdos-Ginzburg-Ziv type generalization of a known result concerning a monochromatic solution of the above inequality in a 2-coloring of the positive integers.

On zero sum Ramsey numbers: Multiple copies of a graph

As a consequence of our main result, a theorem of Schrijver and Seymour that determines the zero sum Ramsey numbers for the family of all r-hypertrees on m edges and a theorem of Bialostocki and Dierker that determines the zero sum Ramsey numbers for r-hypermatchings are combined into a single theorem. Another consequence is the determination of zero sum Ramsey numbers of multiple copies of some small graphs.

On a Variation of Schur Numbers

Abstract. For every m≥3, let n=R (L3 (m)) be the least integer such that for every 2-coloring of the set S={1, 2, …, n}, there exists in S a monochromatic solution to the following system.¶¶ The main result of this paper is that¶¶ Moreover, it is shown that, up to a switching of the colors, there exists a unique 2-coloring of the set {1, 2, …, R(L3 (m)) −1} that avoids a monochromatic solution to the above system.

On the degree of regularity of some equations

In this paper we investigate the behaviour of the solutions of equations ΣI=1n aixi = b, where Σi=1n, ai = 0 and b ≠ 0, with respect to colorings of the set N of positive integers. It turns out that for any b ≠ 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 − x2 = x2 − x1 + b. For this equation, for b odd and 2-colorings, only an odd-even coloring prevents a monochromatic solution. For b even and 2-colorings, always monochromatic solutions can be found, and bounds for the corresponding Rado numbers are given. If one imposes the ordering x1 < x2 < x3, then there exists already a 4-coloring of N, which prevents a monochromatic solution of x3 − x2 = x2 − x1 + b, where b ϵ N.

A generalization of rotational tournaments

Abstract The notions of rotational tournament and the associated symbol set are generalized to r -tournaments. It is shown that a necessary and sufficient condition for the existence of a rotational r -tournament on n vertices is ( n , r )=1. A scheme to generate rotational r - tournaments is given, along with some examples.

Generalization of some Ramsey-type theorems for matchings

Abstract For a graph G let RM( G ) be the smallest integer R, if it exists, such that every coloring of the edges of K R by an arbitrary number of colors implies a subgraph of K R isomorphic to G that is either monochromatic or has the property that no two of its edges have the same color. We generalize the theorem r ( nK 2 , n −1)= n 2 − n +2, where r ( nK 2 , n −1) is the Ramsey number for n matchings in an ( n −1)-coloring of the complete graph. Namely, we prove that RM( nK 2 )= r ( nK 2 , n −1)= n 2 − n +2. In addition, we generalize the theorem r ( nK 2 ,2)=3 n −1 by considering colorings with three and five colors. Several further possible generalizations for hypermatchings in hypergraphs are suggested.