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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
 

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
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
j g - 1. Let G(k, 2) denote the smallest positive integer g such that every nearly consecutive

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics