An Improved Construction of Progression-Free Sets

Abstract

The problem of constructing dense subsets S of {1, 2, . . . , n} that contain no three-term arithmetic progression was introduced by Erdős and Turán in 1936. They have presented a construction with |S| = Ω(n log 3 2 ) elements. Their construction was improved by Salem and Spencer, and further improved by Behrend in 1946. The lower bound of Behrend is

An improved construction of progression-free sets

“…Hence by (14), the overall number N of integer lattice points in C ∩ S that do not belong to Ext(B) (and thus, do not belong to Ext(C ∩ S), because S ⊆ B) satisfies…”

“…Hence the overall running time is at most n·2 O( √ log n) . (See also [14] for an improved analysis of the running time. The improved analysis yields the estimate n 2 Ω( √ log n) , i.e., sublinear in n but superlinear in the size of the returned set.…”

“…The preliminary version of this paper was published in the electronic archive [14] in January 2008. Since then several authors continued our line of research.…”

“…In an even more recent development O'Bryant [26] has combined our techniques with those of Rankin [27] and Green and Wolf [20], and improved Rankin's lower bound by a factor log ǫ n, for some small positive ǫ = ǫ(k). (In the preliminary version of our paper [14] we anticipated that our techniques could be useful to improve Rankin's bound.) We believe that similar ideas may help improving some of the results of [29,31,1,2,3,10].…”

Sunflowers and Testing Triangle-Freeness of Functions

“…Whereas H-freeness is known to be strongly testable for every graph H, it turns out that the graph H significantly affects the dependence of the query complexity on ε. Alon proved in [1] that for every bipartite graph H, the one-sided error query complexity of testing H-freeness is polynomial in 1/ε, and that for every non-bipartite graph H, it is superpolynomial in 1/ε, namely, at least (1/ε) Ω(log(1/ε)) . Interestingly, the lower bound for the non-bipartite case relies on a construction of Behrend [7] of dense sets of integers with no 3-term arithmetic progressions (see also [32,16]) and on an extension of this construction given in [1].…”