Infinite combinatorics and definability

Algebra and Infinite Combinatorics

Infinite Asymptotic Combinatorics

The following combinatorial theorems, some of which were known for every finite n in all infinite structures, are proved in ZFC for every infinite cardinal ν in all sufficiently large structures. (a) A new extension of Miller’s theorem [18]. (b) An upper bound of ρ on the list-conflict-free number of ρ-uniform families of sets which satisfy C(ρ, ν) for cardinals ν and ρ ≥ iω(n). (c) An upper bo...

Infinite Combinatorics: from Finite to Infinite

We investigate the relationship between some theorems in finite combinatorics and their infinite counterparts: given a “finite” result how one can get an “infinite” version of it? We will also analyze the relationship between the proofs of a “finite” theorem and the proof of its “infinite” version. Besides these comparisons, the paper gives a proof of a theorem of Erdős, Grünwald and Vázsonyi g...

Elementary submodels in infinite combinatorics

We show that usage of elementary submodels is a simple but powerful method to prove theorems, or to simplify proofs in infinite combinatorics. First we introduce all the necessary concepts of logic, then we prove classical theorems using elementary submodels. We also present a new proof of Nash-Williams’s theorem on cycle-decomposition of graphs, and finally we obtain some new decomposition the...

Permitted trigonometric thin sets and infinite combinatorics

We investigate properties of permitted trigonometric thin sets and construct uncountable permitted sets under some set-theoretical assumptions.