A new proof of the density Hales-Jewett theorem

Polymath, D. H. J.

doi:10.4007/annals.2012.175.3.6

Cited by 70 publications

(8 citation statements)

References 31 publications

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“…In 2009, inspired in part by his previous work on Szemerédi's theorem, Gowers proposed to locate a purely combinatorial proof of the density Hales-Jewett theorem by a creative new paradigm-an online crowdsourced effort, where dozens of mathematicians, mostly communicating through blogs and wikis, would contribute and debate possible attack strategies. After a very intensive sevenweek effort involving thousands of comments by many mathematicians, such a combinatorial proof was finally obtained in 2010, with the results eventually being published in [Pol12] under the pseudonym "D.H.J. Polymath;" the initials here stand for "Density Hales Jewett."…”

Section: Terence Taomentioning

confidence: 99%

Robert Israel “Bob” Jewett (1937–2022)

Bloom¹,

Gardner²,

Hales³

et al. 2023

Notices Amer. Math. Soc.

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his mother, Mame (Mary) née Katz, was born in Providence to parents from Russia. In 1946, Bob's family, including his older sister Rosalie, moved to Venice, California, a relocation partially motivated by Bob's problems with hay fever. Bob's much younger brother Phil was born in 1948.Bob attended the local public schools and was very interested, while at Venice High School, not only in physics and mathematics, but also animals and insects. In 1955, he started college at Caltech. There was no zoology major, so he focused on physics and mathematics, eventually the latter. In 1958, he received Honorable Mention for his individual performance in the nationwide Putnam competition in mathematics, helping the Caltech team to place third in the nation. Bob also stood out as a volleyball and track and field star.Bob graduated from Caltech in 1959 and went to the University of Oregon for graduate work in mathematics. Before this move, however, he worked for the summer at Caltech's Jet Propulsion Laboratory (JPL), in the coding theory section headed by Solomon Golomb. He was also employed there in the summers of 1960 and 1961. During that time he wrote several research papers, one containing the combinatorial Hales-Jewett theorem [HJ63].Bob received his PhD in 1963, with a thesis on analysis on locally compact abelian groups written under the Communicated by Notices Associate Editor Emilie Purvine.

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Section: Terence Taomentioning

confidence: 99%

Robert Israel “Bob” Jewett (1937–2022)

Bloom¹,

Gardner²,

Hales³

et al. 2023

Notices Amer. Math. Soc.

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“…Previously, the best known bound was O(1/ log * n), achieved by appealing to a quantitative version of the density Hales-Jewett theorem [33]. Theorem 1.8 makes progress on a question of Green [23] and on a question of Haszla, Holenstein and Mossel [30].…”

Section: Extremal Combinatoricsmentioning

confidence: 99%

On approximability of satisfiable k -CSPs: I

Bhangale

Khot

Minzer

2022

Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing

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We prove a stability result for general 3-wise correlations over distributions satisfying mild connectivity properties. More concretely, we show that if Σ, Γ and Φ are alphabets of constant size, and µ is a distribution over Σ × Γ × Φ satisfying: (1) the probability of each atom is at least Ω(1), (2) µ is pairwise connected, and (3) µ has no Abelian embeddings into (Z, +), then the following holds. Any triplets ofmust arise from an Abelian group associated with the distribution µ. More specifically, we show that there is an Abelian group (H, +) of constant size such that for any such f, g and h, the function f (and similarly g and h) is correlated with a function of the form f (x) = χ(σ(x 1 ), . . . , σ(x n ))L(x), where σ : Σ → H is some map, χ ∈ Ĥ⊗n is a character, and L : Σ n → C is a low-degree function with bounded 2-norm.En route we prove a few additional results that may be of independent interest, such as an improved direct product theorem, as well as a result we refer to as a "restriction inverse theorem" about the structure of functions that, under random restrictions, with noticeable probability have significant correlation with a product function.In companion papers, we show applications of our results to the fields of Probabilistically Checkable Proofs, as well as various areas in discrete mathematics such as extremal combinatorics and additive combinatorics.

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“…Whereas Citizen Science focuses on the fact that contributors are not professional scientists, this aspect is less central in Crowd Science: Although most Crowd Science projects engage with citizen crowds, crowds may also consist of professional scientists and other experts. Examples include Harvard Medical School's effort to crowdsource research problems from within the university community (Guinan, Boudreau, and Lakhani 2013), as well as a series of Polymath projects that require high levels of training in mathematics (Polymath 2012). Discussions of Crowd Science typically start from the vantage point of professional scientists who enlist crowds to advance their research.…”

Section: Crowd Sciencementioning

confidence: 99%

“…Given that crowd members are not socialised in the traditional institution of science and their number is large, there is also a limited role for authorship and peer recognition, although a few CS projects have granted authorship to individual crowd members or have recognised the crowd under a collective pseudonym (e.g. Polymath 2012;Khatib et al 2011;Lintott et al 2009). As such, CS projects appear to rely on a more limited range of motivators: One is crowd members' intrinsic interest in particular topics such as astronomy or ornithology (Sauermann and Franzoni 2013).…”

Section: Provision Of Rewardsmentioning

confidence: 99%

Crowds, citizens, and science: a multi-dimensional framework and agenda for future research

Franzoni

Poetz

Sauermann

2021

Industry and Innovation

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Research projects that actively involve 'crowds' or non-professional 'citizen scientists' are attracting growing attention. Such projects promise to increase scientific productivity while also connecting science with the general public. We make three contributions. First, we argue that the largely separate literatures on 'Crowd Science' and 'Citizen Science' investigate strongly overlapping sets of projects but take different disciplinary lenses. Closer integration can enrich research on Crowd and Citizen Science (CS). Second, we propose a framework to profile projects with respect to four types of crowd contributions: activities, knowledge, resources, and decisions. This framework also accommodates machines and algorithms, which increasingly complement or replace professional and non-professional researchers as a third actor. Finally, we outline a research agenda anchored on important underlying organisational challenges of CS projects. This agenda can advance our understanding of Crowd and Citizen Science, yield practical recommendations for project design, and contribute to the broader organisational literature.

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A new proof of the density Hales-Jewett theorem

Cited by 70 publications

References 31 publications

Robert Israel “Bob” Jewett (1937–2022)

Robert Israel “Bob” Jewett (1937–2022)

On approximability of satisfiable k -CSPs: I

Crowds, citizens, and science: a multi-dimensional framework and agenda for future research

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