81 research outputs found
Davies-trees in infinite combinatorics
This short note, prepared for the Logic Colloquium 2014, provides an
introduction to Davies-trees and presents new applications in infinite
combinatorics. In particular, we give new and simple proofs to the following
theorems of P. Komj\'ath: every -almost disjoint family of sets is
essentially disjoint for any ; is the union of
clouds if the continuum is at most for any ;
every uncountably chromatic graph contains -connected uncountably chromatic
subgraphs for every .Comment: 8 pages, prepared for the Logic Colloquium 201
Universal graphs with forbidden subgraphs and algebraic closure
We apply model theoretic methods to the problem of existence of countable
universal graphs with finitely many forbidden connected subgraphs. We show that
to a large extent the question reduces to one of local finiteness of an
associated''algebraic closure'' operator. The main applications are new
examples of universal graphs with forbidden subgraphs and simplified treatments
of some previously known cases
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