# A tail cone version of the Halpern-L\"auchli theorem at a large cardinal

## TL;DR

This paper proves a consistency result for a tail cone version of the Halpern-L"auchli theorem at large cardinals, extending classical combinatorial partition results to higher set-theoretic contexts.

## Contribution

It introduces a tail cone variant of the Halpern-L"auchli theorem at large cardinals and demonstrates its consistency, broadening the scope of partition theorems in set theory.

## Key findings

- Established the consistency of the tail cone Halpern-L"auchli theorem at large cardinals.
- Generalized polarized partition relations from rational numbers to larger saturated linear orders.
- Extended classical combinatorial theorems to the context of large cardinals.

## Abstract

The classical Halpern-L\"auchli theorem states that for any finite coloring of a finite product of finitely branching perfect trees of height $\omega$, there exist strong subtrees sharing the same level set such that tuples consisting of elements lying on the same level get the same color. Relative to large cardinals, we establish the consistency of a tail cone version of the Halpern-L\"auchli theorem at large cardinal, which, roughly speaking, deals with many colorings simultaneously and diagonally. Among other applications, we generalize a polarized partition relation on rational numbers due to Laver and Galvin to one on linear orders of larger saturation.