On the degree of regularity of some equations

✍ Scribed by Arie Bialostocki; Hanno Lefmann; Terry Meerdink

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 530 KB
• Volume: 150
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(95)00174-u

✦ Synopsis

In this paper we investigate the behaviour of the solutions of equations ~i=t aix, = b, where ~--~=~ ai = 0 and b # 0, with respect to colorings of the set N of positive integers. It tunas out that for any b # 0 there exists an 8-coloring of N, admitting no monochromatic solution of x3 -x2 = x2 -xl + 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 xj < x2 < x3, then there exists already a 4-coloring of ~1, which prevents a monochromatic solution of x3 -x2 = x2 -xl + b, where b E ~.