Problems
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R_dih(P_a^alt, K_b) = 1 + (a-1)(b-1) for all a ≥ 4, b ≥ 1 — the a ≥ 4 slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817). Combined with the a = 3 case (see sibling entry), this resolves Conjecture 4.9 in full for a ≥ 3.
Fulek defined a weight-five three-row 0-1 matrix L_3 and asked whether ex(n, L_3) = O(n). It is: every r × s matrix avoiding L_3 has at most 27r + 2s ones, so 6n - 8 ≤ ex(n,L_3) ≤ 29n for n ≥ 5. The same argument covers an infinite family…
Improved lower bounds for nine classical Ramsey numbers, including R(3,13) ≥ 61, R(3,18) ≥ 100, and seven R(4,k) records up to R(4,20) ≥ 237, found by AlphaEvolve-discovered search algorithms.
Is the sequence W_0, W_1, …, W_n counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
VibeMathed records no statement for this problem. See erdosproblems.com for the original.
R_dih(P_3^alt, K_b) = R_cyc(P_3^alt, K_b) = 2b - 1 for all b ∈ N — the a = 3 slice of Conjecture 4.9 (Damnjanović–Đorđević, arXiv:2607.06817) and Conjecture 4.23 (Bašić–Damnjanović–Stevanović–Stošić, arXiv:2604.16188).
Mason conjectured the following: let M be a matroid of rank r, and let W_i denote the number of flats of M of rank i. Is it true that for all 1 ≤ i ≤ r - 1, we have W_i^2 ≥ W_i + 1W_i - 1? This is false; a counterexample is given by a…
The directed five-dimensional torus D_5(m) has a Hamilton decomposition for every odd m ≥ 3, extending the decomposition program for directed tori beyond the three-dimensional case.
Monical, Tokcan and Yong conjectured that every fixed positive power of the Vandermonde determinant fails to have saturated Newton polytope in sufficiently many variables. For every even power k ≥ 4 there is an explicit lattice point of…
There exists a Hadamard matrix of order 668: a matrix H∈-1,1^668×668 such that HH^ T=668I_668. Equivalently, the 668 rows of H are pairwise orthogonal.
How large can a Bruhat interval in S_n that is a poset hypercube be? Using a permutation pattern suggested by AlphaEvolve, the authors exhibit hypercube intervals of dimension O(n log n) for n a power of 2, matching the largest possible…
Subbarao and Verma asked in 1999 (Problem 5.7, first part) whether the complementary Bell numbers f(n) = B_n(-1) take any given value only finitely many times. Campbell proves they do: for every fixed integer the fiber is finite, a result…
Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.
Erdos and Szemeredi conjectured that every finite set of reals satisfies max(|A+A|,|AA|) ≥ |A|^2-o(1). False: there are arbitrarily large A ⊂ R, of algebraic integers in a number field of degree asymp log|A|, with max(|A+A|,|AA|) ≤ |A|^2-c…
A problem from Fajtlowicz's Graffiti program, studied by Erdős and Staton, on the Havel-Hakimi residue of common-divisor graphs. The paper resolves the problem and extends it, determining the residue's first-order scale and its nontrivial…
For A ⊂ F_p of density 1/2, call A almost affine invariant under φ(x) = ax+b if |A triangle φ(A)| = o(p). Problem 90 asks for the threshold K below which A can be almost affine invariant simultaneously under all such φ with |a|, |b| ≤ K…
Tuza conjectured that every finite simple graph satisfies τ(G) ≤ 2ν(G), where ν counts pairwise edge-disjoint triangles and τ is the fewest edges whose deletion leaves the graph triangle-free. Puleo had proved it for maximum average degree…
Let T_k be the least t such that every equinumerous t-coloring of [tn] contains a rainbow k-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured T_k = Θ(k^2); Conlon, Fox and Sudakov proved T_k = O(k^2 log…
Akbari, Alikhani, Oboudi and Peng conjectured in 2010 that 0 and -2 are the only integer roots of the domination polynomial D(G, x), proven for trees and unicyclic graphs and verified exhaustively for small orders. The paper gives a…
For a sequence of n distinct reals, determine the largest constant c such that some monotonic subsequence always has sum exceeding (c-o(1))·(1/√n) times the total sum. Resolved as c = 1.