Progressions in Sequences of Nearly Consecutive Integers

For k > 2 and r ≥ 2, let G(k, r) denote the smallest positive integer g such that every increasing sequence of g integers {a1, a2, . . . , ag} with gaps aj+1 − aj ∈ {1, . . . , r}, 1 ≤ j ≤ g − 1 contains a k-term arithmetic progression. Brown and Hare [4] proved that G(k, 2) > (k − 1)/2(4 3 )(k−1)/2 and that G(k, 2s−1) > (sk−2 /ek)(1+o(1)) for all s ≥ 2. Here we improve these bounds and prove that G(k, 2) > 2k−O( √ k) and, more generally, that for every fixed r ≥ 2 there exists a constant cr > 0 such that G(k, r) > rk−cr √ k for all k. A sequence of integers {a1, a2, . . . , ag} is called nearly consecutive if aj+1 − aj ∈ {1, 2} for 1 ≤

• Type: Journal
• Year: 1998
• Where: JCT
• Authors: Noga Alon, Ayal Zaks
• Keywords: Increasing Sequence, JCT 1998, K-term Arithmetic Progression, Smallest Positive Integer