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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…

Let f_3(N) be the least size forcing a set A ⊆ 1,…,N to contain distinct a,b,c with a+b, a+c and b+c all in A. The upper bound f_3(N) ≤ 5N/8 + O(1) matches the standard construction [N/8,N/4] ∪ [N/2,N], so f_3(N) = 5N/8 + O(1).

Let p be a complex polynomial of degree n≥2 whose zeros all lie in the closed unit disk. For every zero a of p, there is a critical point ζ satisfying |ζ-a|<1, except when |a|=1 and p is a nonzero scalar multiple of z^n-a^n.

Does the Kannan-Lovász-Simonovits variance inequality hold with a universal constant for every quadratic form of an isotropic log-concave random vector - that is, is Var⟨ MX, X⟩ ≤ C E|∇⟨ MX, X⟩|^2 for every symmetric M?

Swinnerton-Dyer (1981) proved R-equivalence trivial on smooth cubic surfaces over p-adic fields with good reduction, except for three special types. The paper resolves two long-standing exceptional cases: triviality for the diagonal cubic…

Given planks of fixed total width, how should they be placed to cover as much of a convex body's volume as possible? Karoly Bezdek asked whether, for a Euclidean ball, the optimum is a single plank centred at the origin. It is, and the…

A cyclic meander induces a cyclic permutation on its 2n marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.

For an inclusion-free hypergraph on n vertices, a weight assignment w:[n]→[d] is isolating when a unique edge attains minimum weight. Faber and Harris conjectured that the number of isolating assignments is at least n∑_j=0^d-1 j^n-1,…

Norine conjectured that every red-blue edge-colouring of the n-dimensional hypercube Q_n in which antipodal edges get opposite colours contains a monochromatic path from some vertex to its antipode. The paper proves it, via a chain-level…

Arjevani et al. asked whether almost-surely bounded oracle error permits a better rate than bounded variance for smooth nonconvex stochastic optimization. It does not: every randomized adaptive algorithm still needs Omega(dL/eps^2 + dL…

Gill introduced the probabilistic automatic complexity A_P(w) of a string: the least number of states of a probabilistic finite automaton for which w is the unique most probably accepted string of its length. He asked whether A_P is…

Erdos asked whether a finite unit-distance graph in the plane can have independence ratio below 1/4. One exists, built on the geometric fractional chromatic number framework of Matolcsi, Ruzsa, Varga and Zsamboki plus a carefully chosen…

Localizing Bernstein theory to prove lower bounds for the Lebesgue constants of Lagrange interpolation, with application to a problem of Erdős and Turán and to a conjectured bound from the interpolation literature.

Let A(x) count n ≤ x such that every prime p | n has a divisor d > 1 of n with d ≡ 1 pmod p. Erdos asked whether A(x)/x = exp(-(c+o(1))√log xloglog x). It does, with c = 1/(2√log 2).