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Estimate the least excess g_k(N) forcing k integers whose pairwise sums all lie in a dense subset of 1, …, 2N; in particular, determine the positive variant h_4(n).

Han and Xiong extended the Gaussian binomial coefficient to positive rational index and conjectured that its integer trace, the integer-exponent part of the resulting power series, is coefficientwise largest at the integer point. Ono's…

Let N(k, ℓ) be the least N such that every f : [N] → -1, 1 has a k-term arithmetic progression P with |∑_n ∈ P f(n)| ≥ ℓ. In particular, is N(k, 2) ≤ C^k?

For irreducible covering sets of size k, determine their count, the possible largest modulus, the maximal reciprocal sum, and whether divisor-set examples occur infinitely often.

For a transcendental entire function, how fast can |f(z)| be forced to grow along a path to infinity, and how short can such a path be in terms of the maximum modulus M(r, f)?

Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. Their conjectured lower bound is established for…

Shokurov's global index conjecture, in the setting of foliations. Proved for foliations in dimension at most three, which also answers a question of Liu, Meng and Xie in dimension three.

The classical Abbott-Hanson recurrence gives S(k+2) ≥ 9S(k)+4 for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted S-templates, a more flexible form of Rowley's template construction, yield S(k+2) ≥…

The Hessian conjecture HC_n asks whether every polynomial f with det Hess(f) ∈ C^× has a polynomial gradient inverse. It is known for n ≤ 3, false for n ≥ 5, and open exactly in dimension four, where it implies the plane Jacobian…

Erdős, Herzog and Piranian (1958) asked whether the regular n-gon maximizes the product of pairwise distances among n points of fixed diameter. After the recent discovery that it does not for even n, this paper proves the first exponential…

For A=0,1,2^d, write f(d) for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether f(d)≥ 2^d for all d, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on f(d).

If origin-symmetric convex bodies K, L ⊂ R^n satisfy vol_m(K ∩ E) ≤ vol_m(L ∩ E) for every m-dimensional subspace E with 1 < m < n, does vol_n(K) ≤ vol_n(L) follow? Answered affirmatively for subspace dimensions m = 2 and m = 3.

Is it true that for every ε,η>0 there exists a k such that the density of n for which P(n(n+1)…(n+k))>n^1-ε is at least 1-η, where P(m) is the greatest prime divisor of m? A short argument via the Matomäki-Radziwiłł theorem establishes the…

Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for…

The arithmetic Kakeya conjecture asserts the infimum of sum-difference exponents is 1, which would imply the Kakeya conjecture in all dimensions. In the bounded-slope-count regime, Tao establishes that the exponents converge to 2 at a rate…

Fuglede's conjecture asks whether a set tiles exactly when it is spectral. The paper proves it for an infinite sequence of square-free order cyclic groups: the tile-to-spectral direction for all square-free cyclic groups, and the…

A structured special case of the Matrix Spencer conjecture, reached through the representation theory of finite-dimensional C*-algebras: the conjectured discrepancy bound holds for every family of contractions contained in a suitable…

Hall and Ho conjectured how the zeros of the heat-flow-evolved characteristic polynomial of a random matrix behave in the large-n limit. General cases are proved; in particular, for a complex Ginibre matrix the empirical measure of those…

Borsuk's conjecture asked whether every bounded set in R^n can be partitioned into n+1 subsets of smaller diameter. It is false in dimension 63: there is a set of 321 points in R^63 whose smaller-diameter subsets have at most 5 points, so…

What is the minimum asymptotic density δ_k of monochromatic k-term arithmetic progressions in every two-colouring of 1, …, n? The exact certificate gives δ_3 = 117/2192, matching the known 548-bead colouring.