Halpern–Läuchli theorem explained

In mathematics, the Halpern–Läuchli theorem is a partition result about finite products of infinite trees. Its original purpose was to give a model for set theory in which the Boolean prime ideal theorem is true but the axiom of choice is false. It is often called the Halpern–Läuchli theorem, but the proper attribution for the theorem as it is formulated below is to Halpern–Läuchli–Laver–Pincus or HLLP (named after James D. Halpern, Hans Läuchli, Richard Laver, and David Pincus), following .

Let d,r < ω,

\langleT_i:i\ind\rangle

be a sequence of finitely splitting trees of height ω. Let

cup_n\left(\prod_i<dT_i(n)\right)=C₁\cup … \cupC_r,

then there exists a sequence of subtrees

\langleS_i:i\ind\rangle

strongly embedded in

\langleT_i:i\ind\rangle

such that

cup_n\left(\prod_i<dS_i(n)\right)\subsetC_kforsomek\ler.

Alternatively, let

	d
S
	\langleT_i:i\ind\rangle

=cup_n\left(\prod_i<dT_i(n)\right)

and

	d=cup
S
	\langleT_i:i\ind\rangle

	d
S
	\langleT_i:i\ind\rangle

The HLLP theorem says that not only is the collection

S^d

partition regular for each d < ω, but that the homogeneous subtree guaranteed by the theorem is strongly embedded in

T=\langleT_i:i\ind\rangle.