Some combinatorics involving ξ-large sets

Formalization of some central theorems in combinatorics of finite sets

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth’s decomposition theorem, Mirsky’s theorem, Hall’s marriage theorem and the Erdős-Szekeres theorem. Dilworth’s decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mi...

Combinatorics of sets

1. Let F be a collection of subsets A1, A2, . . . of {1, . . . , n}, such that for each i 6= j, Ai ∩Aj 6= ∅. Prove that F has size at most 2n−1. Solution: For each set S ∈ 2, observe that at most one of S and S is contained in F . 2. Suppose that F above has size exactly 2n−1. Must there be a common element x ∈ {1, . . . , n} which is contained by every Ai? Solution: No. First, observe that in ...

Combinatorics of lopsided sets

We develop a theory of isometric subgraphs of hypercubes for which a certain inheritance of isometry plays a crucial role. It is well known that median graphs and closely related graphs embedded in hypercubes bear geometric features that involve realizations by solid cubical complexes or are expressed by Euler-type counting formulae for cubical faces. Such properties can also be established for...

Some small large sets of t-designs

Partitioning Α–large Sets: Some Lower Bounds

Let α → (β)m denote the property: if A is an α–large set of natural numbers and [A]r is partitioned into m parts, then there exists a β– large subset of A which is homogeneous for this partition. Here the notion of largeness is in the sense of the so–called Hardy hierarchy. We give a lower bound for α in terms of β,m, r for some specific β. This paper is a continuation of our work [2] and [3] o...