Green 2026 100 open problems

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Ben Green, 100 open problems. problem list (author's manuscript, circulated since 2018; PDF compiled January 2026). No notice is printed in the file; the author's page that links it (https://people.maths.ox.ac.uk/greenbj/) states no terms; the term is unstated.

Green's personal list of 100 open problems, circulated since 2018 and kept updated, spans twelve sections, among them sumsets and bases, Sidon sets, and sieving. Each entry carries a short commentary with references and dated Update paragraphs recording partial or complete solutions - for example Problem 49, the polynomial Freiman-Ruzsa (Marton) conjecture over F_2^n, is marked solved with the Gowers-Manners-Tao-Green proof, while Problem 50, polynomial Bogolyubov, records Kosciuszko's n - O(log^(3+eta)(1/alpha)) bound. Numbering is deliberately frozen so solved entries keep their slot, and the author points readers to the erdosproblems.com discussion forums. For problem 1192 this was a negative check: searching the sumsets-and-bases section and the whole text turns up no just-basis formulation and no Ruzsa just-basis mention, so the list offers no order-r >= 3 restatement of that question.

Source: https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf.

Bears on. #687: Problem 46, p. 23 (PDF p. 23 of the January 2026 PDF read for this card, text layer): "What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p\pmod p$, one for each prime $p\le x$?", the problem's $Y(x)$; the comments call it "the Jacobsthal problem", record the lower bound $y\gg x\log x\log\log\log x/\log\log x$ from [121], any improvement of which would enlarge the known lower bound on the largest prime gap, name Iwaniec's $y\ll x^2$ [176] as the best upper bound, and conjecture $y\ll x^{1+o(1)}$, a proof of which, the author notes, would not improve the upper bound on prime gaps but only bound what one method of producing them can achieve; the problem's two displayed questions. #689: Problem 45, p. 23 (PDF p. 23, text layer): "Can we pick residue classes $a_p\pmod p$, one for each prime $p\le N$, such that every integer $\le N$ lies in at least 10 of them?", traced by the comments to [109, Section 6, Problem 6] (Erdős's 1980 survey), where, the comments report, Erdős says he cannot answer it with $10$ replaced by $2$; the problem's question; "Update 2025" points to the site's page for the problem. #1202: Problem 44, p. 22 (PDF p. 22 of the January 2026 PDF read for this card, page image and text layer), a qualified row: "Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2\le p_1<p_2<\cdots<p_{1000}<N^{9/10}$. Does the remaining set have size at most $\frac1{10}N$?"; the comments (pp. 22--23) trace it to [109, Section 6, Problem 3] (Erdős's 1980 survey), report Erdős's remark that the large sieve gives an affirmative answer when every prime is below $N^{1/2}$, and add that the author knows of nothing on it beyond Erdős's survey of nearly forty years earlier, whose treatment of it has apparently not been cited since. Problem 44 fixes the parameters ($1000$ primes, exponent $9/10$, bound $N/10$) where the problem's statement quantifies $\epsilon$, $\eta$ and $k$; it is a cousin of the problem, not its question, and the list records no solution. #1192: a negative check; the list has no just-basis formulation and does not restate the problem's question for any order $r$ (Section 3, Sumsets and bases, and the whole text searched).

Results to transcribe.

• Problem 49 (Solved): Polynomial Freiman-Ruzsa / Marton conjecture in F_2^n: a set with |A+A| <= K|A| is covered by K^O(1) translates of a subspace of size at most |A|; solved by Gowers, Manners, Tao and Green, with the integer analog still open.
• Problem 50: Polynomial Bogolyubov over F_2^n: does 10A contain a coset of a subspace of dimension n - O(log(1/alpha))? Best known is n - O(log^(4+o(1))(1/alpha)) (Sanders), with Kosciuszko obtaining n - O(log^(3+eta)(1/alpha)) for mA - mA.
• Problem 51: For A in F_2^n of density alpha, what is the largest coset guaranteed inside 2A? Known: dimension >>_alpha n (at least c(alpha) n for some c(alpha) > 0 depending on alpha), but not always n - sqrt(n).
• Structure of the list: Twelve thematic sections with frozen numbering, commentary and dated Update paragraphs; no just-basis problem appears and nothing restates Problem 1192's question, which is the relevant negative finding here; the one order-r basis question, the Erdos-Sarkozy-Sos bonus question on p. 19 whether an infinite Sidon set can be an asymptotic basis of order 3, is marked resolved in the affirmative by Pilatte (Update 2023).

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.