Call a set A ⊆ R paradoxical if there are disjoint A0, A1 ⊆ A such that both A0 and A1 are equidecomposable with A via countabbly many translations. X ⊆ R is hereditarily nonparadoxical if no uncountable subset of X is paradoxical. Penconek raised the question if every hereditarily nonparadoxical set X ⊆ R is the union of countably many sets, each omitting nontrivial solutions of x − y = z − t....

A continuum means comp6lct, connected metric space. A hereditarily unicoherent and arcwise connected continuum is called a dendroid. It follows that it is hereditarily decomposable ([2], (47), p. 239). A hereditarily unicoherent and hereditarily decomposable continuum is said to be a A.-dendroid. Thus, every dendroid is a }.-dendroid and an arcwise connected A.-dendroid is a dendroid. Note that...

A graph G is hereditarily dominated by a class D of connected graphs if each connected induced subgraph of G contains a dominating induced subgraph belonging to D. In this paper we characterize graphs hereditarily dominated by classes of complete bipartite graphs, stars, connected bipartite graphs, and complete k-partite graphs.

Research Article Guram Bezhanishvili, Nick Bezhanishvili, Joel Lucero-Bryan and Jan van Mill S4.3 and hereditarily extremally disconnected spaces Abstract: Themodal logic S4.3 de nes the class of hereditarily extremally disconnected spaces (HED-spaces). We construct a countable HED-subspaceX of the Gleason cover of the real closed unit interval [0, 1] such that S4.3 is the logic ofX.