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problems
Let n_k be the least integer greater than 2k for which ∏_i=1^k (n_k - i) has no prime factor in (k, 2k). How rapidly must n_k grow?
A Banach space is primary if in every decomposition into two complemented subspaces one summand is isomorphic to the whole. Lechner, Motakis, Müller and Schlumprecht identified the primariness of L_p(L_1) as a prominent remaining open…
The realisation problem asks which unital Banach algebras arise as the Calkin algebra B(X)/K(X) of some Banach space. Recorded in Tarbard's thesis and studied by Horváth and Kania. The paper exhibits a unital Banach algebra that cannot be…
Vinzant conjectured, in a form later restated by Bandeira, that the 4M-4 threshold for injective complex phase retrieval is sharp. Part (1) holds: for A ∈ C^N × M with N = 4M-5 and i.i.d. standard complex Gaussian entries, the phase…
Must every sufficiently large node set admit bounded labels that force any polynomial fitting almost all labels at degree below (1+ε)n to have arbitrarily large uniform norm? Claimed via Beurling density for Bernstein spaces.
Let μ be a probability measure on the unit circle with Verblunsky coefficients α. Lukic conjectured that a weighted entropy condition with finitely many critical points is equivalent to a decomposition of α into components localized at…
How well can an arbitrary boolean constraint satisfaction problem of arity k be approximated in polynomial time? The paper gives a (k/2^k)-approximation, improving the previous best constant of 0.626612 k/2^k due to Makarychev and…
Steurer conjectured in 2010 that any family of n unit vectors with polynomially small average correlation E_i,j|⟨ v_i,v_j⟩| ≤ n^-ε contains linear-sized constant-separated sets. Refuted in a strong sense, using sparse high-dimensional…
Erdos and Graham asked whether a positive-density subset of 1,…,N can avoid having any two distinct elements a,b whose unit fractions average to a unit fraction. It can: there is a constant c>0 such that for all large N some A ⊆ 1,…,N of…
The unrestricted planar Berenstein conjecture holds that overdetermined Dirichlet-Neumann data characterize the disc. Disproved: a bounded simply connected domain with real-analytic Jordan boundary that is not a disc, carrying a nonzero…
Given online vectors v_t ∈ R^d with |v_t|_2 ≤ 1, can signs ε_t ∈ -1, 1 be chosen in O(dT) total time so that every prefix has ℓ_∞ discrepancy O(√log T) with high probability? The previous optimal algorithm ran in time exponential in T and…
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly t red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
Banks and Martin conjectured in 2013 that for a primitive set A and any set Q of primes, the Erdos sum of the members of A composed only of primes in Q is at most the corresponding sum over Q itself. The unrestricted form turned out to be…
Ziegler proved every simplicial d-dimensional 0/1-polytope has at most 2d vertices, and asked whether attaining 2d vertices forces central symmetry (i.e. a 0/1 cross-polytope). Known true for d ≤ 6; open since ~2000.
Is the irreversibility of entanglement manipulation robust in the strong-converse sense - a strict separation between the exponential strong-converse distillable entanglement and the entanglement cost, as conjectured by Lami and Regula?…
Lions asked whether the variational solution of a non-autonomous divergence-form problem has maximal L2-regularity under Holder continuity in time of the coefficients. Disproved at the half-Holder endpoint: a bounded, uniformly elliptic,…
A graph G on n vertices with k edges is t-edge-balanced if every graph on n vertices with t edges is contained in exactly the same number of subgraphs of K_n isomorphic to G. Infinite families were known for t = 2, but no example was known…
Mauri and Moraga posed a two-part question about log Calabi-Yau pairs whose boundary decomposes into big divisors. Both parts have negative answers.
How large can a measurable A ⊆ [0,R]^2 be while avoiding the vertices of upward-oriented axis-aligned right triangles of area 1/2? At most O_c(R^2/(log R)^c), with a matching-shaped lower bound construction.