Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9515365 | Journal of Combinatorial Theory, Series A | 2005 | 14 Pages |
Abstract
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,â¦,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⩾1and b⩾1, the disjunctive Rado number for the equations x1+a=x2 and x1+b=x2 is a+b+1-gcd(a,b) if agcd(a,b)+bgcd(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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Brenda Johnson, Daniel Schaal,