Problems

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)

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 =…

For integers n, r and a prime p satisfying r < p, we have ex(n, K_r, C_≥ p^prime) ≤ n-1/p-2 binomp-1r. Equality holds only for connected n-vertex graphs consisting of n-1/p-2 maximal 2-connected blocks each isomorphic to K_p-1.

For n≥2 B(K_2,n,Z_2n)≤4n-3 .

Note that the divisibility conditions in (22) should be equivalent to those in (23) if a t-(n,k,λ) exists. It is open if they are equivalent.

We further conjecture that an elliptic quadric is incident with m modulo q points of an m-ovoid of Q(4, q).

For m, r nonnegative integers, P_(m^r)(q,t)=∑_substackλ ⊂(m^r) 2-core(λ)=0f_λ(q^-m,q^r/T;q,t)tildeK_(m^r)-λ(q,t,T;± q^1/2,± t^1/2) and for m, r, n nonnegative integers such that r \leq n, P_(m^r)(x^±;q,t)=∑_substackλ ⊂(m^r)…

Let m ≥ 3 be an integer, and let Γ be a graph coprime to K_m such that Aut(Γ) ≠ 1. If Γ is connected and R-thin, then (Γ, K_m) is stable.