Problems

For an extension G = A rtimes B of elementary abelian p-groups with a ∈ A satisfying C_B(a) = 1, must H = ⟨ a, B⟩ satisfy rank(Z(H) ∩ H') ≤ rank(B)? An explicit extension violates the bound.

Bounds the weighted sum ∑ 1/(a log a) taken over primitive sets of integers (sets where no element divides another).

Can the critical-exponent relation a + b = 1 at the jamming transition, observed numerically to high precision in the full replica-symmetry-breaking solution of hard spheres, be derived analytically from the scaling equations?

Is every group sofic - does every group admit approximate finite permutation representations? A central open question of geometric group theory since Gromov introduced soficity: soficity implies Gottschalk's surjunctivity conjecture,…

What is the largest A⊆1,…,N such that all subset sums ∑_n∈ S1/n (over S⊆ A) are distinct?

For a single-source unsplittable flow, find the optimal universal additive constant C s.t. every feasible fractional flow x with arc costs c should admit an unsplittable routing y with c^top y ≤ c^top x and y_a ≤ x_a + C · d_max on every…

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

Does there exist a group with more than one but only finitely many maximal locally soluble normal subgroups? An explicit group with exactly two settles it.

For every finite set A⊂ Z with |A|≥ 2, define C(A)=log(|A+A|/|A|)/log(|A-A|/|A|). Determine the largest possible value of C(A), equivalently the least universal exponent c such that |A+A|/|A| ≤ (|A-A|/|A|)^c for every such set A. The…

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.

For four particles and local dimension D ≥ 4, can a complete edge-coloured, complex-weighted graph have unit perfect-matching amplitude for every monochromatic inherited colouring and zero for every nonmonochromatic one? Ruled out for the…

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

How large must arithmetic circuits and formulas computing the n × n permanent be? New lower bounds include an arithmetic-formula bound of order n^4/log n, far beyond the quadratic barrier that stood for decades.

For a finite abelian group G, let Φ(G) be the absolutely convex hull of the specified trilinear kernels and Φ'(G) its restriction where the third factor depends only on x_1 + x_2. Is Φ(G) = Φ'(G)? A counterexample over Z/3Z separates the…

How dense can a sphere packing in R^n be as n → ∞? The Kabatiansky-Levenshtein upper bound stood for almost fifty years; the new proof improves the asymptotic upper bound all the way down to the Cohn-Elkies linear-programming threshold.

Let A=1≤ a_1< a_2<… be a set of integers such that Abackslash B is complete for any finite subset B and not complete for any infinite subset B. If a_n+1/a_n ≥ 1+ε for all n, must lim_n a_n+1/a_n=(1+√5)/2? Under the reading where the ratio…

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

Does the value of a two-player quantum game decay exponentially under parallel repetition, as Raz's theorem gives for classical games? Yes: an exponential parallel repetition theorem holds for arbitrary finite two-player quantum games.

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