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Gao, Huo and Ma asked whether for every fixed k≥3 there is a function fk(n)→∞ such that every n-vertex (k+1)-critical graph contains fk(n) consecutive cycle lengths. The paper settles this and two related problems on cycle lengths and cycles with chords under chromatic and degree constraints.
As Tao notes on the problem page, the claim establishes natural LOWER density at least 1-eta but not that the natural density exists, so the problem as stated remains technically open
Is the density of Chernoff's distribution - the law of argmaxt{W(t)−t2} for two-sided Brownian motion W - strongly log-concave, as conjectured by Balabdaoui and Wellner in 2014?
Energy measures of any two nonconstant harmonic functions on the standard Sierpinski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence that the Radon-Nikodym densities are Lp-integrable for 1<p<log15/log9. That range is confirmed: the associated quantities are uniformly bounded for arbitrary ordered pairs of nonconstant harmonic functions.
Can the k-distinct language - words over [n] of length at most k with no repeated symbol - be recognized by an acyclic NFA of size cknO(1) for some c<4? A construction of size 21.96992knO(1)<3.918knO(1) answers yes.
Around the Kemeny median problem, which stays open for m=3 and m=5 voters, the paper refutes three conjectures on tournament inducibility: both conjectures of Milosz, Hamel and Pierrot (the 3-cycle extension for odd m≥5, and FAS=HS3 at n=11), and Shepard's threshold conjecture.
A graph G is maximal non-Hamiltonian if it is non-Hamiltonian but G+e is Hamiltonian for every nonedge e. In 1994 Vu Dinh Hoa conjectured a property of G−V(C) for a longest cycle C of such a graph. Disproved by an explicit base graph on 56 vertices, extended to larger orders.
A cyclic meander induces a cyclic permutation on its 2n marked intersection points. Schwartz's conjecture on the quadratic growth of the associated meander number is resolved.
A dimension-independent subgaussian concentration bound for Gaussian vectors under coordinate-wise nonlinear maps, valid for any bounded function under a well-conditioned covariance, which answers a question of Simone Bombari on sign quantization.
What is the optimal uniform continuity bound for quantum conditional entropy in trace distance, depending only on the dimension of the conditioned system? The sharp bound h2(δ)+δlog(d2−1) up to δ=1−d−2, conjectured by Wilde, is proved.
Two open problems about extracting order from trees in real-valued functions. A quantitative function analogue of Hodges's tree-to-order extraction yields an at most double-exponential bound on dual sequential fat-shattering dimension, resolving the first. A new proof of Daskalakis-Golowich tight-threshold extraction, avoiding multicolored Ramsey numbers, resolves the second, which concerned repairing the bound in a…
Chafai, Dadoun and Youssef asked whether the quadratically penalised logarithmic energy of mean empirical spectral distributions is monotone in the dimension, for Wigner matrices and for matrices with i.i.d. entries. Neither holds: a finite-energy Wigner counterexample and a one-parameter family of Gaussian-regularised Bernoulli entry laws answer both questions negatively.
an escape path dominating every power of |z| with bounded initial length is constructed, and universal positive-power lower bounds are ruled out; the broader variant remains open
The key step is a lower bound on the truncated imbalance sum, which yields every Erdos-Gallai inequality for the sorted imbalance list; a parity computation finishes it.
Sabok asked whether the compact convex set S′(X) attached to a separable metric space of diameter at most one is always a simplex, and whether S′(U1) is the Poulsen simplex. Both answers are negative, with obstructions already visible for finite X and for the Urysohn space.
Zhi-Wei Sun conjectured a closed evaluation of a truncated Legendre-symbol determinant. For every prime p≡3(mod4) it equals ⌊(p−2)/3⌋2x, proved by reducing to inverse data for Chapman's full Legendre-symbol matrix and evaluating that with Vsemirnov's factorization and a Schur-Pfaffian resolvent identity.
Does there exist an integer polynomial f of degree at least two and a set A⊆Z such that every integer has a unique representation n=a+f(k)? A manuscript claims the thirteenth powers admit a tiling complement.
Every finite simple connected graph G with
∣V(G)∣=2d+1,diam(G)=d≥3
satisfies
W(G)≤W(C2d+1)=2(2d+1)d(d+1).
The claimed equality cases are exactly C2d+1 for every d≥3, the double star D2,3 when d=3, and the nine-vertex tree T1,2,2=S(2,3,3) when d=4.
Let H(n) be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than n. With k1=1 and kn=⌊n/2⌋+k⌊n/2⌋+k⌈n/2⌉, prove H(n)≥ckn for some constant c>1, already for n=15, with a constructive algorithm.
Given online vectors vt∈Rd with ∥vt∥2≤1, can signs εt∈{−1,1} be chosen in O(dT) total time so that every prefix has ℓ∞ discrepancy O(logT) with high probability? The previous optimal algorithm ran in time exponential in T and d.
Douglas and Yang attach to each nonzero vector x of a quasinilpotent operator T a local resolvent-growth exponent kx, giving the power set Λ(T)={kx:x=0}. Ji and Zhang asked whether 1 always belongs to Λ(T). It does, for every quasinilpotent operator on every Banach space. Moreover Λ(T)=[0,1] for every backward unilateral weighted shift on ℓp with strictly decreasin…
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 finitely generated groups. They do not.
Curto et al. (Advances in Applied Mathematics, 2024) conjectured that every stable fixed point of a threshold-linear network is minimal. Disproved: an explicit competitive 3-neuron TLN has a stable fixed point whose support strictly contains another's, and 3 neurons is proven smallest possible.
Every minimally generically globally rigid graph in Rd containing a subgraph isomorphic to Kd+2 is itself isomorphic to Kd+2, confirming Conjecture 6.3 of Garamvölgyi, Jackson and Jordán (2025).
In the Frankl-Pach-Erdős circle of VC-dimension problems, the first arXiv version of the paper posed the k=3 case of a witness construction question. ChatGPT 5.4 Pro answered it; the published construction generalizes the model's response, and the conversation transcript is public.