Extremal problems for affine cubes of integers - University of Manitoba

6.4 Final step in proof of lower bound for h(d, r)
We now prove for d ≥ 2, and r ≥ 2 that r 2d−1 (1−o(1)) d ≤ h(d, r), where o(1)
tends to zero as r increases.
Proof of lower bound in Theorem 2.5: Fix n large (large enough to apply
Theorem 6.2). It suffices to provide a partition of [1, n] into at most n d+o(1)
2d−1 parts, each part affine d-cube-free. By Theorem 6.2, let [1, n] = X1 ∪ . . . ∪ Xq
be a partition into at most q = e3√ln n parts, each Xi not containing any three
term arithmetic progression, and each |Xi| ≤ n/2 √ ln n . For each i = 1, . . . q,
by Lemma 6.4, partition Xi = Ai,1 ∪ . . . ∪ Ai,k(i),
k(i) ≤ 12
2 d
2 d −1
|Xi|
d
2d−1 < 12
2 d
2 d −1
n/2 √ ln n d
2 d −1
into parts, none of which contains a replete affine d-cube. For each i =
1, . . . , q and for each j = 1, . . . , k(i), Ai,j contains neither a three term arithmetic
progression nor a replete affine d-cube, so by Lemma 6.1, each Ai,j is
affine d-cube-free. The total number of partitions is at most
q · max k(i) ≤ e 3√ ln n ·
= n d+o(1)
2 d −1 ,
12
2 d
2d (n/2
−1 − 1
√ d
ln n
) 2d−1 For large n. ✷
We conclude by noting that we have proved Theorem 2.5 for d ≥ 2,
however this is only an improvement over known results for d ≥ 3 (cf [2]).
Acknowledgements. We thank the referee for suggestions leading to improvements
in both results and presentation. We particularly appreciated
comments regarding quantification.
References
[1] F. A. Behrend, On sets of integers which contain no three elements
in arithmetic progression, Proc. Nat. Acad. Sci. 32 (1946), 331-332.
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