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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 loglog F(x)/log x → 1, i.e. F(x) = exp(x^1+o(1)).

In the all-heads coin game a player starts with n coins, each showing heads with probability p; each round all remaining coins are flipped, the player must set aside at least one head (losing if none shows), and wins once all coins are set…

VibeMathed records no statement for this problem. See erdosproblems.com for the original.

Let L^nf be the Lagrange interpolation polynomials of a continuous f on the Chebyshev nodes. Prove that, for any closed A⊆ [-1,1], there exists a continuous function f such that A is the set of limit points of L^nf(x).

Let G be a simple connected graph on n≥ 5 vertices. If the maximum over all vertices v of ℓ(v) - the independence number of the subgraph induced by the open neighborhood N(v) - is at most 1, must G be well totally dominated? Answered…

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.

If g_3(n) is the largest size of A ⊆ [1,n] with fewer than three representations of every product a_1 a_2, does its conjectured second-order normalized term converge? The candidate proof gives an explicit limit constant.

Does the conjectured universal formula for normalized alternating syzygy power sums of numerical semigroup rings hold for every index?

For every connected graph G, is α(G) ≤ ⌊ b(G) - log(ecc_avg(G)) ⌋, where b(G) is the largest induced-bipartite-subgraph order? An 11-vertex counterexample - a triangle with four leaves on each of two vertices - has α = 9 against bound 8.

For odd k with gcd(n,k) = gcd(n+1,k) = 1, is N_k(n) ≡ ⌊ (k+1)/4 ⌋ pmod 2, where N_k(n) counts pairs 1 ≤ b_i ≤ (k-1)/2 with b_1 + b_2 ≥ (k+1)/2 and b_2 ≡ n b_1 pmod k? Conjectured by Chen and Gendron; its proof removes a conditional step in…

If the power graph of a finite group contains no induced path on four vertices, must it also contain no induced cycle of length at least four - that is, is every cograph power graph chordal?

What is the maximum size of a binary code of given minimum distance? The linear-programming bounds of McEliece, Rodemich, Rumsey and Welch (1977) resisted improvement for half a century. The new upper bounds are exponentially stronger at…

Does the sum of the reciprocals of all primitive pseudoperfect numbers converge?

For an infinite planar set in strong general position, how large can the chromatic and clique numbers of its positive-integer-distance graph be - in particular, can the chromatic number be infinite? Yes: there is such a set, no three…

If each integer has at most r representations m = pa with p prime and a ∈ A ⊆ [1, N], what is the best upper bound for ∑_a ∈ A 1/a? The candidate proof gives the matching order Θ_r(log N / loglog N).

For the least cutoff c(n) after which every k occurs as the number of homothetic cubes in a decomposition of the unit n-cube, is c(n) ≫ n^n? The Lean proof shows c(n) = o(n^n) along odd dimensions.

Determine the Shannon capacities of odd cycles beyond C_5, or improve the best explicit bounds. Lovasz's theta function settled C_5 in 1979 and every longer odd cycle has stayed open since. The current records, all obtained with model…

Let a,b,c>1 be pairwise coprime integers. Is every large integer a sum of distinct numbers of the form a^k b^l c^m (k,l,m≥ 0), none dividing another?

VibeMathed records no statement for this problem. See erdosproblems.com for the original.