 
Summary: Progressions in Sequences of Nearly Consecutive Integers
Noga Alon
Ayal Zaks
Abstract
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
kterm arithmetic progression. Brown and Hare [4] proved that G(k, 2) > (k  1)/2(4
3 )(k1)/2
and that G(k, 2s1) > (sk2
/ek)(1+o(1)) for all s 2. Here we improve these bounds and prove
that G(k, 2) > 2kO(
k)
and, more generally, that for every fixed r 2 there exists a constant
cr > 0 such that G(k, r) > rkcr
k
for all k.
A sequence of integers {a1, a2, . . . , ag} is called nearly consecutive if aj+1  aj {1, 2} for 1
j g  1. Let G(k, 2) denote the smallest positive integer g such that every nearly consecutive
