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Estimate the number F(x) of minimal distinct covering systems whose moduli all lie in [1,x]. The candidate proof gives loglogF(x)/logx→1, i.e. F(x)=exp(x1+o(1)).
Let A(x) count n≤x such that every prime p∣n has a divisor d>1 of n with d≡1(modp). Erdos asked whether A(x)/x=exp(−(c+o(1))logxloglogx). It does, with c=1/(2log2).
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.
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.
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.
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 is nondecreasing and then nonincreasing.
The problem statement is ambiguous: the limit-exists reading is claimed proved (Lean), while the convergence-from-hypotheses reading was disproved by a Lean-checked construction of Price that the community classes as a variant
exact largest modulus 3·2^{k-3} for k ≥ 5, near-linear least maximum, reciprocal mass Θ(log k), and an infinite divisor family; the counting asymptotic rests on the cited BBMST theorem
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.
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…
Donner proved in 1992 that the list color function Pℓ(G,k) equals the chromatic polynomial P(G,k) once k is large. Kaul and Mudrock asked whether the analogue holds for Hanlon's unlabeled chromatic polynomial, and could not settle even the edgeless graph, which they posed as a conjecture. The conjecture is true, and it implies that a disconnected graph satisfies the unlabeled analogue of Donner's result wh…
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.
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.
Answers the obstruction rather than the whole question: it says exactly when the Matrix-Tree Theorem can be applied and classifies the chambers, and gives a compact five-graviton formula outside the decay region. Simplifying the general solution, which is what the source paper left to future work, remains open.
Proves Haglund's Conjecture 4 for k=1: every non-real first-quadrant zero of Φ1+tΦ2 is simple with strictly decreasing imaginary part, no branch escapes forward, and every finite-multiplicity real collision stays real afterwards. The cases k≥2 remain open. Two readings worth separating: Conjecture 4 asserts the monotone descent alone, so the no-escape and stays-real statements are this paper's own ad…
The dihedral case only, for every a≥4 and b≥1; the substance is the upper bound, which the source paper's own computations could not reach. Together with the sibling a = 3 entry this proves Conjecture 4.9's claim 1+(a−1)(b−1) for all a≥3; the conjecture's trivial a = 1, 2 cases are unaddressed by either entry, and the cyclic analogue Rcyc(Paalt,Kb) for a≥4 remains open. The engin…
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).
Twenty-eight individual exact values, each proved by SAT certificate (coloring at n-1, UNSAT at n). The synthesis theorem covers every c >= 2 whose c+1 is divisible by 3, 4, 5, or 7 (~66% of integers). The prime-reduction corollary shows the full conjecture (R(c)=40c+41 for all c >= 2) is equivalent to checking primes p >= 89; all primes through 83 are settled. What stays open: the conjecture at c=88 (p=89) and ever…
Pavez-Signe (2024) conjectured a Dirac-type condition for spanning H-subdivisions and asked whether the subdivision paths can additionally be required to have similar lengths; Lee (2025) resolved the existence conjecture in the stronger digraph setting. Answered affirmatively with epsilon-room: for every ε>0 there is C0 such that every n-vertex digraph D with n≥C0h and minimum semi-deg…
Nazarov conjectured that for s∈(1,3/2) the quadratic form of the spectral fractional Dirichlet Laplacian strictly increases under u↦∣u∣ when u changes sign. Proved and substantially generalized, with the same conclusion for the restricted form.
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.
Aldroubi, Cabrelli, Krishtal and Molter conjectured that for a bounded normal operator T and any vector g, the normalized orbit {Tkg/∥Tkg∥:k≥0} is never a frame. It can be: an explicit construction produces a normalized orbit that is a frame.