Problems

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

With N = 2^k - 1 and wt(n) the binary Hamming weight, Tu and Deng conjectured that for every 1 ≤ t ≤ N-1 at most 2^k-1 pairs (a,b) satisfy a + b ≡ t pmod N and wt(a) + wt(b) < k. Proved in full.

For f(z) = ∏_i=1^n (z - z_i) with all |z_i| ≤ 1, let ρ(f) be the radius of the largest disc contained in z : |f(z)| < 1. Is ρ(f) ≫ 1/n? The worst case is now known to be Θ(1/n), with the explicit bound ρ(f) ≥ (log 2)/n.

Give an explicit profinite presentation of Gal(overlineQ_2 / Q_2). The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-2…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Whether there are infinitely many integers a, b, n with a, b ≥ ε n such that a!· b! divides n!·(a+b-n)! while a+b exceeds n by more than C·log n.

Is the maximum size of a set A⊆ 1,…,N such that ab+1 is never squarefree (for all a,b∈ A) achieved by taking those n≡ 7pmod25? Resolved for all sufficiently large N: any near-maximal A is contained in n≡ 7pmod25 or n≡ 18pmod25, leaving…

Let F(n) be the largest A⊆1,…,n with anmid bc for distinct a,b,c∈ A. Is F(n)=π(n)+(C+o(1)) n^2/3(log n)^-2 for some constant C?

Let X_1,…,X_n be independent nonnegative random variables with EX_i ≤ 1, and let S be their sum. Is P(S < ES + 1) ≥ 1/e? Feige proved the constant 1/13 and conjectured the sharp 1/e. Three independent July 2026 proofs settle it, both…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

If W(k) is the least N such that every two-colouring of 1, …, N contains a monochromatic k-term arithmetic progression, must W(k+1) - W(k) → ∞?

Can a nonabelian group admit a Rota-Baxter operator that is surjective but not injective? A construction shows yes.

Can the edges of a finite connected multigraph, given a closed eulerian trail, be partitioned into circuits so that no circuit contains two edges used consecutively in the trail? The proof in fact four-colours the edges to satisfy the…

Is the EMD coupling square a^2 a function of the metric three-jet on an explicit active, non-null, simple-spectrum family of truncated Einstein-Maxwell-dilaton data, and can one more derivative recover it? Proved: no function of the common…

Nathanson asked which subsets of N can occur as product intersection sets of a family of semigroup subsets, for arbitrary and for decreasing families (his Problems 10 and 11). Both are solved by complete classifications.

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Let k,r≥ 2. Does there exist a set A⊆ N that contains no non-trivial arithmetic progression of length k+1, yet in any r-colouring of A there must exist a monochromatic non-trivial arithmetic progression of length k? Answered in the…

Donner proved in 1992 that the list color function P_ℓ(G,k) equals the chromatic polynomial P(G,k) once k is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even…