They also conjectured that for Jki<feâ£o, pk!(n)Spk2(n) and for ko<h<k2i pki(n)<pk2(n). They verify this conjecture for n^32. Recently Todd published a table of all the pkin) for n^lOO, and it is easy to verify the conjecture for w^glOO. I am unable to prove or disprove this conjecture. They also remark that pk0{n) differs from c-inm i0g n by i e s s than 1 for n^ 32; (2) shows that for large n...

Given an r × r complex matrix T , if T = U |T | is the polar decomposition of T , then, the Aluthge transform is defined by ∆ (T ) = |T |U |T |. Let ∆n(T ) denote the n-times iterated Aluthge transform of T , i.e. ∆0(T ) = T and ∆n(T ) = ∆(∆n−1(T )), n ∈ N. We prove that the sequence {∆n(T )}n∈N converges for every r × r matrix T . This result was conjecturated by Jung, Ko and Pearcy in 2003. W...

We propose a homological approach to two conjectures descended from the Erdős-Ko-Rado Theorem, one due to Chvátal and the other to Frankl and Füredi. We apply the method to reprove, and in one case improve, results of these authors related to their conjectures.

We consider Erdős-Ko-Rado sets of generators in classical finite polar spaces. These are sets of generators that all intersect non-trivially. We characterize the Erdős-Ko-Rado sets of generators of maximum size in all polar spaces, except for H(4n+ 1, q) with n ≥ 2.

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdős–Ko–Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corres...

Let k and m be positive integers. A collection of k-multisets from {1, . . . ,m} is intersecting if every pair of multisets from the collection is intersecting. We prove that for m ≥ k +1, the size of the largest such collection is ( m+k−2 k−1 ) and that when m > k + 1, only a collection of all the k-multisets containing a fixed element will attain this bound. The size and structure of the larg...