Problems

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

Let k,r≥ 2. Does there exist a set A⊆ N that contains no non-trivial arithmetic progression of length k+1, yet in any r-colouring of A there must exist a monochromatic non-trivial arithmetic progression of length k? Answered in the…

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.

A monic prime P of F_q[T] is a c-Wieferich prime if ρ_P(1) ≡ 1 bmod P^2 for the Carlitz module ρ. On limited data and proofs in degrees 2 and 3, Thakur suggested in 2015 that in odd characteristic every c-Wieferich prime has degree…

For the least t_k(n) with n | t_k(n)(t_k(n)+1)…(t_k(n)+k-1), do the conjectured logarithmic-saving and adjacent-length estimates hold on average? Both answered affirmatively, with c = 1/2048 admissible in the t_2 bound.

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

Find a nontrivial lower bound or upper bound of QMX(n).

It is conjectured that if division and reciprocation, as well as subtraction and negation, are also made allowed operations, then they are never necessary to achieve a maximizing combination.

Conjecture. If G ∈ G(k,n) and if p= lfloorn/2⌋ orlceiln/2⌉ ifn ≡ 1(bmod 4),k iseven;n ≡ 3(bmod 4),k iseven, lfloorn/2⌋ ifn ≡ 1(bmod 4),k isodd, n-2/2orn+2/2 ifn ≡ 2(bmod 4),k iseven, lceiln/2⌉ ifn ≡ 3(bmod 4),k isodd, then R^′(G)≥…

Let S be the smallest family of subsets of I such that each t-subset of I occurs in at most \lambda blocks.Then S contains all subsets of size \geqslant(n-r') , where r' is the largest integer satisfying ( cn-t n-t )+( cn-t n-t-1 )+...+(…

Conjecture 1.6.2. Let h_i(q) be defined by qh_i(q)=g_i+1,i(q) . Then f_m(b,q)=(1-q)^m-1 +q ∑_i=0^m-1(1-q)^m-y^(i)h_i(q)b^i +∑_i=m^binomm2-1(1-q)^m-y^(i)g_m,i(q)b^i +fracqb^binomm2(m-1)!∑_i=0^m-2⟨ cm-1 i ⟩ q^i.

Given a cancellative, finitely-generated monoid M in which lcm's exist, is M necessarily a Garside monoid? That is, does there exist a Garside element Δ in M ?

The study of the number of edges as well as the chromatic number of the derivative Euler Phi set-graphs (lcm-divisor and lcm-relatively prime) remains open.

Let X be a compact Hausdorff space and T:X → X a continuous map. For any open U ⊆ X and any ℓ ∈N , there exists n ∈N with U ∩ T^-nU ∩ T^-2nU ∩… ∩ T^-ℓ nU ≠∅,or (3) T^-inU ∩ T^-jnU=∅ ∀0 ≤ i<j ≤ℓ. (4)

Provided lower and upper bounds for f(k).

Let ε > 0 be any constant and let q be a sufficiently large prime power. Let L be a set of at least q^5/2+ε lines in F_q^3 such that no plane contains more than (1/2)q^3/2 lines of L. Then, |P(L)| ≥ (1-o(1))q^3.

Find other possible values of the parameter d and the corresponding d-antimagic labeling of type (1, 1, 1) for the hexagonal plane map H_n^m.

Working in the differential tower of groups A imathS with A abelian of order r, when k ≤n the critical group K(V(U^kD^k)_n)=K(Ind_A imathS_n-k^A imathS_n1) is given, as a list of elementary divisors,…

Is it true for every t that ¿ lim_n →∞F(n;t)/n=1/2?

We conjecture that any fixed value occurs finitely many times.