Partition regularity

In <a href="/facts/Combinatorics/k14xUoKT">combinatorics</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, partition regularity is one notion of largeness for a <a href="/facts/Set_system/WbYof7lP">collection</a> of sets.
Given a set 
 
 
 
 X
 
 
 {\displaystyle X}
 
, a collection of <a href="/facts/Subset/SJGERmbA">subsets</a> 
 
 
 
 
 S
 
 ⊂
 
 
 P
 
 
 (
 X
 )
 
 
 {\displaystyle \mathbb {S} \subset {\mathcal {P}}(X)}
 
 is called partition regular if every set A in the collection has the property that, no matter how A is partitioned into finitely many subsets, at least one of the subsets will also belong to the collection. That is,
for any 
 
 
 
 A
 ∈
 
 S
 
 
 
 {\displaystyle A\in \mathbb {S} }
 
, and any finite partition 
 
 
 
 A
 =
 
 C
 
 1
 
 
 ∪
 
 C
 
 2
 
 
 ∪
 ⋯
 ∪
 
 C
 
 n
 
 
 
 
 {\displaystyle A=C_{1}\cup C_{2}\cup \cdots \cup C_{n}}
 
, there exists an i ≤ n such that 
 
 
 
 
 C
 
 i
 
 
 
 
 {\displaystyle C_{i}}
 
 belongs to 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
. <a href="/facts/Ramsey_theory/CzLLM2j8">Ramsey theory</a> is sometimes characterized as the study of which collections 
 
 
 
 
 S
 
 
 
 {\displaystyle \mathbb {S} }
 
 are partition regular.

Partition regularity open-in-new

Partition regularity