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Combinatorics
Green's open problem 18
Suppose that G is a finite group, and let A⊂G×G be a subset of density α.
Is it true that there are ≫α∣G∣3
triples
x,y,g
such that
(x,y),(gx,y),(x,gy)
all lie in
A
?
Note: A is taken as
α
-dense, i.e.
∣A∣≥α∣G∣2
[Au16, Question 2]
$2,680 bounty·1 piece from 1 person·last one 11 days ago
week of 27 Sept 2025 · nothing publishedweek of 4 Oct 2025 · nothing publishedweek of 11 Oct 2025 · nothing publishedweek of 18 Oct 2025 · nothing publishedweek of 25 Oct 2025 · nothing publishedweek of 1 Nov 2025 · nothing published
Solve it
A complete proof or refutation in Lean. The first one the validator accepts takes the whole bounty.
[Au16] Austin, Tim. "Ajtai–Szemerédi theorems over quasirandom groups." Recent trends in combinatorics. Cham: Springer International Publishing, 2016. 453-484.
[So13] Solymosi, Jozsef. "Roth-type theorems in finite groups." European Journal of Combinatorics 34.8 (2013): 1454-1458.
[Go01] Gowers, William T. "A new proof of Szemerédi's theorem." Geometric & Functional Analysis GAFA 11.3 (2001): 465-588.
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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