Arithmetic Progressions in Sequences with Bounded Gaps

✍ Scribed by Tom C Brown; Donovan R Hare

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 528 KB
• Volume: 77
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.1997.2734

✦ Synopsis

Let G(k, r) denote the smallest positive integer g such that if 1=a 1 , a 2 , ..., a g is a strictly increasing sequence of integers with bounded gaps a j+1 &a j r, 1 j g&1, then [a 1 , a 2 , ..., a g ] contains a k-term arithmetic progression.

It is shown that G(k, 2) > -(k & 1)Â2 ( 43 ) (k&1)Â2 , G(k, 3) > (2 k&2 Âek)(1 + o(1)), G(k, 2r&1)>(r k&2 Âek)(1+o(1)), r 2.