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Let (X,B,μ) be a σ -finite measure space and T:X → X a measure preserving transformation. If A ∈B , then there exists n ∈N with μ(A ∩ T^-nA ∩ T^-2nA ∩… ∩ T^-ℓ nA)>0,or (5) μ(T^-inA ∩ T^-jnA)=0 ∀0 ≤ i<j ≤ℓ. (6)

I.e., for hereditary property H, what are the maximal properties H' ⊇ H such that d^*(H') = d^*(H)?

Full characterization/ Examples of strongly minimal k-vertex-rigid graphs in three-space or higher dimensions.

However, it is unknown whether all finite groups are connected 3-CI-groups, and it is conjectured in [15] that the answer is positive.

What is the computational complexity of ~?

If G is a 3-edge-connected plane graph with L(G)=5, then fer(G)=L(G)+1.

What about Wilf equivalence in [m]^* where [m] = 1, 2, …, m? ... Is it true that u ~_m v if and only if u ~ v?

L_2(3),overlineL_2(3),G_0,barG_0 are the only regular graphs for which equality holds in the above inequality.

it is still unknown a connected non-graceful graph that has a vertex relaxed graceful labelling.

For a pr-graph H, the following are equivalent. (i) B_H has no invertible pairs or chains. (ii*) B_H has a parity-symmetric special bipartite-min ordering. (iii) S_H has a semi-conservative WNU-polymorphism.

Do trees of maximum degree at most 4 have exponentially independent sets of linear order?

However, this formula is not particularly useful unless the least integer K such that the MDGP will hold for all k ≥K can be determined. The calculation of this tight bound for a given graph remains an open question.

If all degrees of G are even, then for any two partitions T_1,T_2 of T, the vertices of Q(G,T_1)∩ Q(G,T_2) are T_1-T_2 -feasible, i.e. Q(G,T_1)∩ Q(G,T_2) is the convex hull of T_1-T_2 -feasible vectors.

If F →(nG) , then must F contain at least lfloorr(nG)/|V(G)|⌋ copies of G?

The problem COMPUTECHARBINARY is GapP-complete under many-one reductions.

If the linear compression dimension of S is given by log_{2}(|S|)+o(n) (hence matches the non-linear compression dimension) then S is contained in the union of 2^{o(n)} translates of some subspaces of size at most |S|.

Let K be a clarified formal context with |SRB_k(K)|=n . Then |SOB_k(B(K))|≥ n holds.

This results lead us to establish a conjecture that gives us an algebraic and a combinatorial description of the sandpile group of the cone of the hypercube Q_d of dimension d. More precisely, K(c(Q_d)) ≅ bigoplus_i=1^d Z_2i+1^binomdi =…

(i) For each p ≥ 6, γ_p^(2) > γ_p^(3) and γ_p^(3) < γ_p^(4) < … < γ_p^(p-1). (ii) For each p ≥ 7, Γ_p^(2) > Γ_p^(3) (which is the same as γ_p^(2) > γ_p^(3)), and Γ_p^(3) < Γ_p^(4) < … < Γ_p^(p-1).

For any integer ℓ ≥ 4, there is a constant c_ℓ such that rwsat(n, C_ℓ) = 3/2n + c_ℓ.