We state the dense-set Halpern-Lauchli theorem and sketch Harrington’s proof of the theorem. Harrington’s proof is then adapted to give a new proof of the Boolean Prime Ideal theorem in the first Cohen model. This method is flexible and is shown to admit two natural generalizations. First, it is applied to give a simpler proof of Pincus’ result that ZF+BPI+DC_\kappa does not imply choice. Second, it is applied to make progress towards the goal of proving BPI in iterations of symmetric extensions. Lastly, we show that this proof generalizes Stefanovic's theorem that the Halpern-Lauchli theorem can be derived from BPI in the Cohen model.