Problems
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Is the O'Shea-Zames-Falb multiplier test necessary for robust stability of Lur'e systems with slope-restricted nonlinearities, as conjectured by Carrasco? No: there is a stable Lur'e interconnection, certified by a full-block multiplier,…
For every connected graph G, is α(G) ≤ ⌊ b(G) - log(ecc_avg(G)) ⌋, where b(G) is the largest induced-bipartite-subgraph order? An 11-vertex counterexample - a triangle with four leaves on each of two vertices - has α = 9 against bound 8.
For a graph G, its vertex deck is the multiset of graphs obtained by deleting one vertex. Bowler, Brown, and Fenner (BBF) proposed 2⌊(n−1)/3⌋ as the maximum possible overlap between the decks of two nonisomorphic n-vertex graphs, for all…
Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since…
Albertson and Berman conjectured that for every simple planar graph G on n vertices, the largest vertex set inducing a forest has size at least n/2. The standing lower bound since the same year has been Borodin's 2n/5, from his acyclic…
Lorist and Schwenninger prove Crouzeix's conjecture (arXiv:2608.03841, Lemma 1) by combining a lower bound (their inequality (4)) with an upper bound (inequality (5)). In Remark 2 they observe that (5) alone gives κ ≤ 1 + √1 - ℜ⟨ E_1…
Let U_n⊂ R^n be the difference polytope of a regular n-simplex such that U_n circumscribes a sphere of diameter 1. Then every set of diameter 1 in R^n is covered by a rotated copy of U_n.
Every purely-maximal ideal of a commutative ring is purely-prime, and the converse holds for several important classes of rings; Tarizadeh conjectured (Conjecture 5.8 of his earlier published paper) that in a commutative ring every…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field k of characteristic 0 and any d ≥ 3 there is a d-dimensional…
Ciliberto, Knutsen, Lesieutre, Lozovanu, Miranda, Mustopa and Testa asked a question about effective divisors of positive self-intersection on smooth projective surfaces. The answer is negative, witnessed by a very non-movable effective…
Erdős conjectured, in over a dozen papers spanning 1976 to 1997 and with a $1000 prize attached, that every finite Sidon set extends to a perfect difference set modulo p^2+p+1 for some prime p. Alexeev and Mixon establish that 1,2,4,8 is a…
Jaeger conjectured that every bridgeless cubic graph G admits a Petersen coloring: a map φcolon E(G)→ E(P) into the edges of the Petersen graph P such that, for every vertex v of G, the three edges at v are sent to three edges meeting at a…
Simonovits conjectured that if a forbidden family F with p(F) > 1 has extremal number exceeding the Turan bound by a superlinear surplus, then its extremal graphs are joins of p graphs, each extremal for a family of chromatic number two.…
The BCHM theorem makes the log canonical ring of a projective klt pair finitely generated. Generalized pairs, introduced by Birkar and Zhang, add an auxiliary nef part and have become a central tool in birational geometry, so it is natural…
A group is Howson if the intersection of any two finitely generated subgroups is finitely generated, and strongly Howson if the rank of that intersection is bounded in terms of the two ranks. Zhang asked whether the two coincide for…
Baumslag asked whether a one-relator group G=F/⟨⟨ r⟩⟩ with r a commutator is Hopfian, residually finite or automatic. The paper constructs a family G_m=⟨ a,t | [t,a[a,t]^-m]⟩ answering all three negatively.
A graph on n vertices is very well-covered if every maximal independent set has size n/2. Levit and Mandrescu conjectured that the independence polynomial i(G,x) of every very well-covered graph is unimodal, i.e. its coefficient sequence…
Does a single ReLU neuron trained on modular addition align to one Fourier frequency? Open Problem MAIS-O60, itself posed by Claude Fable 5 under the direction of Lionel Levine, is answered negatively: an explicit construction reaches a…
Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.