A hodgepodge of sets of reals

We open up a grab bag of miscellaneous results and remarks about sets of reals. Results concern: Kysiak and Laver-null sets, Kočinac and γk-sets, Fleissner and square Q-sets, AlikhaniKoopaei and minimal Q-like-sets, Rubin and σ-sets, and Zapletal and the Souslin number. See the survey papers Brown, Cox [1], and Miller [17, 19]. 1 σ-sets are Laver null A subtree T ⊆ ω of the finite sequences of elements of ω = {0, 1, 2, . . .} is called a Laver tree [14] iff there exists s ∈ T (called the root node of T ) with the property that for every t ∈ T with s ⊆ t there are infinitely many n ∈ ω with tn in T . Here tn is the sequence of length exactly one more than t and ending in n. We use [T ] to denote the infinite branches of T , i.e., [T ] = {x ∈ ω : ∀n ∈ ω x n ∈ T} A set X ⊆ ω is Laver-null iff for every Laver tree T there exists a Laver subtree T ′ ⊆ T such that [T ′] ∩X = ∅ This is analogous to the ideal of Marczewski null sets, (s)0. For some background on this topic, see Kysiak and Weiss [12] and Brown [2]. A separable metric space X is a σ-set iff every Gδ in X is also Fσ. It is known to be relatively consistent (Miller [16]) with the usual axioms of set theory that every σ-set is countable. Thanks to the conference organizers: Cosimo Guido, Ljubisa Kočinac, Boaz Tsaban, Liljana Babinkostova, and Marion Scheepers for their generosity in inviting me to speak at the Second Workshop on Coverings, Selections and Games in Topology, December 2005, University of Lecce, Italy. Mathematics Subject Classification 2000: 03E17; 54D20; 03E50