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963 of 1221 statements are formalized in Lean (Formal Conjectures) · 107 carry an Erdős prize · 3 are independent of ZFC and 9 more can't be proved, or can't be disproved, from ZFC alone · “open” counts the computable and Lean-checked ones too.
If \(A \subset \mathbb{N}\) has \(\sum_{n \in A}\frac 1 n = \infty\), then must \(A\) contain arbitrarily long arithmetic progressions? Yes. This was proved by an internal OpenAI model in September 2026 [OAI26]; see the linked Lean proof.
Szemerédi's theorem
**Erdős Problem 139**: Let \(r_k(N)\) be the size of the largest subset of \({1,...,N}\) which does not contain a non-trivial \(k\)-term arithmetic progression. Prove that \(r_k(N) = o(N)\).
Let \(k≥3\). Are there \(k\) consecutive primes in arithmetic progression?
Prove an asymptotic formula for \(r_k(N)\), the largest possible size of a subset of \(\{1, \dots, N\}\) that does not contain any non-trivial \(k\)-term arithmetic progression. That is, find \(f_k\) with \(r_k(N) / f_k(N) \to 1\) as…
On [Mathoverflow](https://mathoverflow.net/a/410815) user [leechlattice](https://mathoverflow.net/users/125498/leechlattice) shows that \(h(n) \ll n^{\frac 2 3}\).
Let \(A=\{a_1,a_2,\ldots\}\subset \mathbb{R}^d\) be an infinite sequence such that \(a_{i+1}-a_i\) is a positive unit vector (i.e. is of the form \((0,0,\ldots,1,0,\ldots,0)\)). For which \(d\) must \(A\) contain a three-term arithmetic…
Let \(k\geq 3\). Must any ordering of \(\mathbb{R}\) contain a monotone \(k\)-term arithmetic progression, that is, some \(x_1 <\cdots < x_k\) which forms an increasing or decreasing \(k\)-term arithmetic progression? The answer is no,…
What is the largest \(k\) such that in any permutation of \(\mathbb{Z}\) there must exist a monotone \(k\)-term arithmetic progression \(x_1 < \cdots < x_k\)? Here a permutation of \(\mathbb{Z}\) is a one-sided arrangement…
Must every permutation of \(\mathbb{N}\) contain a monotone 4-term arithmetic progression? Here a monotone \(k\)-term arithmetic progression in a permutation \(f\) is a set of indices \(i_1 < \cdots < i_k\) such that…
Can \(\mathbb{N}\) be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Let \(V\) be a vector space over the rationals and let \(k\) be a fixed positive integer. Then there is a set \(X_k \subseteq V\) such that \(X_k\) meets every infinite arithmetic progression in \(V\) but \(X_k\) intersects every…
If \(A\subset \mathbb{R}\) does not contain a 3-term arithmetic progression then must \(\mathbb{R}\backslash A\) contain an infinite arithmetic progression? Baumgartner [Ba75] answered this in the negative.
Does the longest arithmetic progression of primes in \(\{1,\ldots,N\}\) have length \(o(\log N)\)?
Green-Tao theorem
Are there arbitrarily long arithmetic progressions of primes? Solution: yes. Ref: Green, Ben and Tao, Terence, _The primes contain arbitrarily long arithmetic progressions_
Stanley sequences
Let \(N\geq 1\). What is the largest \(t\) such that there are \(A_1,\ldots,A_t\subseteq \{1,\ldots,N\}\) with \(A_i\cap A_j\) a non-empty arithmetic progression for all \(i\neq j\)?
If ℕ is \(2\)-coloured then there must exist a monochromatic three-term arithmetic progression \(x,x+d,x+2d\) such that \(d>x\). This was first proved by Brown and Landman [BrLa99], who in fact show that this is always possible with…
Let \(\delta>0\) and \(k\geq 3\). Is it true that there exists \(m\geq 1\) (depending only on \(\delta\) and \(k\)) such that, for all large \(N\), if \(A,B\subseteq \{1,\ldots,N\}\) with \(\lvert A\rvert \geq \delta N\) and…
Let \(\delta_k\) be such that in any \(2\)-colouring of \(\{1,\ldots,n\}\) there exist at least \((\delta_k+o(1))n^2\) many monochromatic \(k\)-term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) for…
Let \(k\geq 3\). Is it true that, in any finite colouring of the integers, there are monochromatic arithmetic progressions of primes of length \(k\)? It follows from the theorem of Green and Tao [GrTa08] (that any set of primes with…