Daniel Schaal

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 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 Generalized Schur Numbers

ABSTRACT Let represent the equation x1 + x2 + ⋅⋅⋅ + xt − 1 = xt. For k ≥ 1, 0 ≤ i ≤ k − 1, and ti ≥ 3, the generalized Schur number S(k; t0, t1, …, tk − 1) is the least positive integer m such that for every k-coloring of {1, 2, …, m}, there exists an i ∈ {0, 1, …, k − 1} such that there exists a solution to that is monochromatic in color i. In this article, we report 26 previously unknown values of S(k; t0, t1, …, tk − 1) and conjecture that for 4 ≤ t0 ≤ t1 ≤ t2, S(3; t0, t1, t2) = t2t1t0 − t2t1 − t2 − 1.

Disjunctive Rado numbers

If L1 and L2 are linear equations, then the disjunctive Rado number of the set {L1, L2} is the least integer n, provided that it exists, such that for every 2-coloring of the set {1, 2 ....,n} there exists a monochromatic solution to either L1 or L2. If such an integer n does not exist, then the disjunctive Rado number is infinite. In this paper, it is shown that for all integers a ≥ 1 and b ≥ 1, the disjunctive Rado number for the equations x1 + a = x2 and x1 + b = x2 is a + b + 1 - gcd(a, b) if a/gcd(a,b) + b/gcd(a,b) is odd and the disjunctive Rado number for these equations is infinite otherwise. It is also shown that for all integers a > 1 and b > 1, the disjunctive Rado number for the equations ax1 = x2 and bx1 = x2 is cs+t-1 if there exist natural numbers c, s, and t such that a = cs and b = ct and s + t is an odd integer and c is the largest such integer, and the disjunctive Rado number for these equations is infinite otherwise.

A note on disjunctive Rado numbers

Abstract For all integers m , n , such that 3⩽ m ⩽ n , let r ( S ( m ), S ( n )) represent the least integer such that for every 2-coloring of the set {1,2,…, r ( S ( m ), S ( n ))} there exists a monochromatic solution to either S ( m ): ∑ i =1 m −1 x i = x m or S ( n ): ∑ i =1 n −1 x i = x n . The integer r ( S ( m ), S ( n )) is called the disjunctive Rado number for the above two equations. In this paper it is determined that r S(m),S(n) = m 2 −m−1 for m⩽n⩽m+1, m 2 −2m+1 for m+2⩽n⩽m 2 −2m+2, n−1 for m 2 −2m+3⩽n⩽m 2 −m−1, m 2 −m−1 for n⩾m 2 −m.

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let

Rado Numbers for the Equation Σ i=1 m-1 x i +c=x m , for Negative Values of c

m\geqslant 3

Two-color Rado numbers for x+y+c=kz

be a positive integer, and let