Problems
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If H is bipartite and r-degenerate, is ex(n;H) ≪ n^2-1/r (a $500 Erdős-Simonovits prize conjecture)? A counterexample refutes the degeneracy conjecture.
Ramachandra and Natarajan conjectured a bound on the pairwise independent correlation gap in their 2025 Operations Research Letters paper. An explicit counterexample refutes it.
For A ⊆ F_p let A^* = (A+A) ∪ (AA). Sárközy conjectured that for all large primes, every set of size at least c√p has A^* = F_p-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together…
How large can the difference between the largest and second-largest distance multiplicities be among n planar points?
Every polynomial map C^n → C^n with constant nonzero Jacobian determinant is invertible, with a polynomial inverse.
The multivariate independence polynomial is the partition function of the hard-core model with per-vertex fugacities. The paper proves a lower bound extending to the multivariate setting a result Tao proved in the univariate case, and…
Teschner conjectured that every finite simple graph G with at least one edge satisfies b(G) ≤ 3/2Δ(G), where b(G) is the bondage number and Δ(G) is the maximum degree. Yavari gives a connected cubic bipartite graph on 18 vertices with…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
For positive integers d and k, let n_k(d) be the maximum order of a graph of maximum degree at most d and diameter at most k. It is shown that lim_d → ∞n_k(d)/d^k = 1 for every fixed k, thereby resolving the asymptotic degree-diameter…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
The Riemann hypothesis asserts that every nontrivial zero of the zeta function lies on the critical line. Short of proving it, the standard measure of progress is the proportion of zeros known unconditionally to lie there: Selberg…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
With N = 2^k - 1 and wt(n) the binary Hamming weight, Tu and Deng conjectured that for every 1 ≤ t ≤ N-1 at most 2^k-1 pairs (a,b) satisfy a + b ≡ t pmod N and wt(a) + wt(b) < k. Proved in full.
For f(z) = ∏_i=1^n (z - z_i) with all |z_i| ≤ 1, let ρ(f) be the radius of the largest disc contained in z : |f(z)| < 1. Is ρ(f) ≫ 1/n? The worst case is now known to be Θ(1/n), with the explicit bound ρ(f) ≥ (log 2)/n.
The lattice triangle problem asks which rational triangles unfold to Veech surfaces; in the hard obtuse window it is conjectured that none do. Via an arithmetic reformulation of the Mirzakhani-Wright rank obstruction, the paper rules out…
If a/b ∈ Q_>0 and b is squarefree, can a/b always be written as a finite sum of reciprocals of distinct products of two distinct primes?
The dimension-five case asks whether, for every nonnegative 5×5 real matrix A whose entries sum to 5, the Dittert functional Φ(A)=∏_i r_i+∏_j c_j-per(A) is uniquely maximized at U_5=J_5/5. The submitted artifact claims the stronger…
Give an explicit profinite presentation of Gal(overlineQ_2 / Q_2). The tame local cases were settled by the early 1980s; the dyadic case was the last one missing. The new presentation has four generators, two word relations and a pro-2…
VibeMathed records no statement for this problem. See erdosproblems.com for the original.