An open workspace for mathematical discovery. Check proofs, make comments, form collaborations.

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.

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.

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.

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…

Find a nontrivial lower bound or upper bound of QMX(n).

Ehrhart equivalence is a necessary and sufficient condition for (not necessarily finite or rational) discrete equidecomposability.

Is it true that for every nonnegative integer k, there exists a connected graph G satisfying φ(G) − κ(G) + 1 = k?

Is there a nice combinatorial proof for the number of interior lattice points of P_n(132,312) ?

We discuss ... including some new ones that we present in this last section (in particular Conjectures 11.3 and 11.5).

φ(n, m) = n(m-1) + 1.

For k a non-negative integer and s, t ≥ k, the coefficient of x^{2s-k}e^{tx} in f_{s+t-1}(x) is given by [x^{2s-k}e^{tx}] f_{s+t-1}(x) = (-1)^{s} \frac{t^{2s+2t-2k-2}}{2^{s-k}\cdot (s-k)! (t-k)!} \cdot Q_{k}(s,t), where Q_{k}(s,t) is a…