The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa $ is strongly compact if and only if the strong tree property holds at $\kappa $, and supercompact if and only if ITP holds at $\kappa $. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph _n$ for all $2 \leq n < \omega $ simultaneously with the strong tree property at $\aleph _{\omega +1}$; we also show that it is consistent for ITP to hold at $\aleph _n$ for all $3 < n < \omega $ and at $\aleph _{\omega +1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.

We show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Chronic idiopathic thrombocytopenic purpura (ITP) is an immune-mediated disorder in which platelets are prematurely destroyed in the reticuloendothelial system by platelet autoantibodies. However, it is becoming clear that the pivotal process of the humoral immune response in the pathogenesis of the disorder is a complex interaction between antigen-presenting cells, T cells and B cells. Furthermore, it is increasingly evident that regulatory T cells play an important role and that T-cell-mediated cytotoxicity contributes to the destruction of platelets in ITP. Different new approaches to immunotherapy in chronic ITP have been explored, including use of anti-CD20, anti-CD154 and anti-CD52 antibodies. So far, these therapies have been antigen-nonspecific and the risk of general immunosuppression is a concern. Thus, improving our understanding of the interaction and relative contribution of humoral and cell-mediated mechanisms is essential for developing antigen-specific immunotherapies for the treatment of this disorder. This review aims to elucidate the current status of knowledge of the cellular and humoral immune components of chronic ITP, together with the implications of this knowledge for therapy.