Problems

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.

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.

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)

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.

Does Theorem 1 hold with no restriction on K? If not, what is the least information needed on K?

For an arbitrary planar graph G, is there a proper grid drawing of G in a grid of polynomial size?

Given a p × q integer matrix M with p ≥ 2, if none of the differences between two rows of M is parallel to 1^{T} , then m(M,n)=(2+o(1))n/log_{p}n.

In PG(3, q) and PG(4, q), the upper bounds (1.5), (1.6) hold for all q.

Let M(n) denote the absolute value of the Möbius function μ[1,W_n]=μ[1,M_n] . Then for n>50 we have M(2n)=n^2⇔ n+1 isprimeandn ≡ 0(bmod6)M(2n)=n^2-1 ⇔ n+1 isprimeandn ≡ 4(bmod6)M(2n+1)=n^2-n ⇔ n+1 isprimeandn ≡ 0(bmod6)M(2n+1)=n^2-n-1 ⇔…

It remains open whether enumeration on chordal graphs can be improved further, so we hereby pose it as an open problem, or whether one can obtain a higher lower bound, which might also be a gap-improvement on general graphs.

Conjecture 18. The top subsegment of the n-core (n=12,14,16, ...) is μ_n/4=μ_n-4(A-3× 2^n-7)⊕ μ_(n-2)/4(A)^3 , where A=M_n-1+2^n-3 is the senior term of the n-core.

Let f(x_1,…,x_n) be a polynomial over a field F given by (1.1) and (1.2). Provided n≥ k, for any finite subset A of F we have |(f(x_1,…,x_n):x_1,…,x_n∈ A, and x_i≠ x_j if i≠ j)| ≥ minp(F)-llbracket n=2 a_1=-a_2rrbracket,…

Specifically, we conjecture that similar processes will work for cycle pendant stars up to stars of size 15.