Citation Context

...at have not used Fourier analysis have generally been accompanied by tower-type bounds, establishing only that r3(N) ≪ N/ log ∗ N; see [25, Chapter 10] for references, as well as the more recent work =-=[17]-=-. (The iterated logarithm of N, log ∗ N, is defined to be the number of times it is necessary to take the logarithm of N in order to get a number less than or equal to 1, and thus grows extremely slow...

Citation Context

...ne progress towards an understanding of the problem.” In this strict sense, Polymath1 was successful well beyond Gowers’s expectations. To date there are two journal articles that have been submitted =-=[26, 25]-=-. Additionally, at least two new proofs of the theorem, plus new bounds on Hales-Jewett numbers were discovered. In this work we will examine whether and to what extent Polymath1 was successful in the...

Citation Context

... to be lower bounded by a very large constant, albeit a constant — specifically it is some sort of Ackerman function of some polynomial in q (in the improved proof of the density Hales-Jewett theorem =-=[Pol09]-=-). Nevertheless it does imply that if log N1(q, d) is sufficiently large as a function of q (a constant we label λq,4), then this allows to conclude that q such “near-parallel” hyperplanes do exist. N...

Citation Context

...nalytic argument. Recently, a combinatorial proof of the density Hales-Jewett theorem, which is an extension of the multidimensional Szemerédi theorem, was discovered as part of the polymath project =-=[81]-=-. As well as its implications in number theory, the removal lemma and its extensions are central to the area of computer science known as property testing. In this area, one would like to find fast al...

Citation Context

...of mathematicians recently conducted an online collaborative analysis of a mathematical problem, it took them longer to write the publishable proof than it did to figure out the approach and solve it =-=[4; 6]-=-. Bwang posted the narrative shown in Figure 3 to the Spring Fest wiki. We then move on to understand Team C's formula for summing up the total # of sticks in n-level diamond. We first tried to used t...

Citation Context

...rogressions [14] and its multidimensional version [5]. Because of its significance the density Hales–Jewett Theorem has received considerable attention and there are, by now, several different proofs =-=[2, 10, 15]-=-. Our goal in this paper is to give yet another proof of the density Hales–Jewett Theorem that is modeled after Polymath’s proof [10] but places one of its crucial parts in a general conceptual framew...

Citation Context

...s consequences, most notably the famous Szemerédi Theorem on arithmetic progressions [36] and its multidimensional version [13]. The best known upper bounds for the numbers DHJ(k, δ) are obtained in =-=[28]-=- and have an Ackermann-type dependence with respect to k. 1We recall that if w = (wi) n−1 i=0 is a variable word over k of length n, then its wildcard set is defined to be the set { i ∈ {0, ..., n− 1}...