On strong chains of uncountable functions

Projections in Weakly Compactly Generated Banach Spaces and Chang's Conjecture

“…In this paper we are interested in the first pair of the form (κ, κ ++ ) that is (ω, ω 2 ) and later in the pairs (ω, λ) for λ > ω 2 . This growth of the gap between cardinal invariants of the structures and the logical and combinatorial phenomena which accompany it, is a classical theme in infinitary combinatorics and the results we obtain follow the general pattern for mathematical structures other than Banach spaces (see for example [3], [9], [10], [18], [19], [22]). It is related to Ramsey theoretic properties of the infinite cardinals.…”

More results in polychromatic Ramsey theory

“…The remedy, which again is due to Todorčević [9], is to use side conditions which are finite matrices of models. This idea has been used by several authors [6,5], and will be a key point in the proof of Theorem 2.…”

“…The following definition abstracts certain properties of the families of models used as side conditions in papers by Todorčević [9], Shelah and Zapletal [6] and Koszmider [5]. We have in mind particularly the side conditions for the forcing P D in [9, section 4], the matrices of models in [6,Definition 18], and the side conditions for the forcing P in [5, Section 3].…”

Proper forcing remastered

“…In §3 we show how to add a chain of length ω 2 in the structure (ω ω 1 1 , < fin ). This result is originally due to Koszmider [5]. Finally, in §4 we give another proof of a result of Baumgartner and Shelah [2] by using side condition forcing to add a thin very tall superatomic Boolean algebra.…”

“…In [5] Koszmider constructed a forcing notion which preserves cardinals and adds an ω 2 chain in (ω ω 1 1 , < fin ). The construction uses an (ω 1 , 1)-morass which is a stationary coding set and is quite involved.…”