Problems

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.

To replace this conjecture, a new conjecture is formulated that we coin as the Brualdi-Li tally conjecture.

Let G' be a graph obtained from a (Δ,1) -bidegreed graph G by inserting a bouquet in an edge of a bouquet internal path. Then (i) if ρ(G)<1+√Δ-1, then ρ(G^′)>ρ(G); (ii) if ρ(G)>1+√Δ-1, then ρ(G^′)<ρ(G); (iii) if ρ(G)=1+√Δ-1, then…

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

it is not known whether \alpha_{\lambda}(k;p) has unimodal coefficients for all \lambda and k.

The element tildew_b is maximal in the weak order on tildeW/W among all dominant elements tildew ∈ tildeW/W : tildew^-1(0) ∈ S(b).

Let Y and Z be t-cross-intersecting sets in G_n whose sizes meet the bound in Theorem 1.2. If n is sufficiently large compared to t, then Y = Z and Y or Y^T is a t-coset.

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

For each D ∈ D and ψ=ψ_D , there is a unique prime factor Z_ψ(x) of P_ψ(x) such that degZ_ψ equals the number of elements in G ψ .