Problems

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…

Which finite triple systems occur in every triple system of uncountable chromatic number? The claimed characterization: exactly those that, after removing isolated vertices, are linear, have every hyperedge-node of their Levi graph meeting…

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 N(k, ℓ) be the least N such that every f : [N] → -1, 1 has a k-term arithmetic progression P with |∑_n ∈ P f(n)| ≥ ℓ. In particular, is N(k, 2) ≤ C^k?

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…

For a finite forbidden triple system G, what exact uncountable chromatic cardinalities occur among G-free triple systems, and how do those spectra interact? The revised manuscript answers the three exact-cardinal questions and claims a…

For irreducible covering sets of size k, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

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…

A monic prime P of F_q[T] is a c-Wieferich prime if ρ_P(1) ≡ 1 bmod P^2 for the Carlitz module ρ. On limited data and proofs in degrees 2 and 3, Thakur suggested in 2015 that in odd characteristic every c-Wieferich prime has degree…

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).

For the least t_k(n) with n | t_k(n)(t_k(n)+1)…(t_k(n)+k-1), do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with c = 1/2048 admissible in the t_2 bound.

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.

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

Bajpai, Dona and Nitsche left three degree-six symplectic hypergeometric monodromy groups unclassified as arithmetic or thin. Two of the three, C-47 and C-55, are arithmetic.

For every finite family F of graphs, is there a single G ∈ F with ex(n;G) ≪_F ex(n;F)? A counterexample refutes the Erdős-Simonovits compactness conjecture.

On the basis of experiments up to 5000 nodes, Papamanthou and Tollis conjectured a relation between the longest paths produced by their MaxSTN and MinSTN algorithms for st-orientations of biconnected graphs. A counterexample refutes it.

If a smooth bounded domain in R^n admits a Neumann eigenfunction of the Laplacian that is constant on the boundary, must the domain be a ball? Pompeiu posed an equivalent integral-equation form in 1929; Schiffer's 1957 reformulation via…