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In this talk I will present a proof of the theorem due to Stevo Todorčević, following a recent exposition by Spencer Unger, which was explained to me by Honza Hubička. If time permits, I will also show how the Halpern–Läuchli theorem implies Milliken's tree theorem, a Ramsey-style theorem in which Halpern–Läuchli plays the role of the pigeonhole principle.
Past Talks
2026/3/18: A short proof of the Halpern–Läuchli theorem
Given at: Seminar on Reckoning
Abstract
A classical product Ramsey theorem states that for every $n$ there exists $N$ such that every $2$-coloring of the edges of $K_{N,N}$ contains a monochromatic copy of $K_{n,n}$. The most direct infinite analogue fails: one can easily construct a $2$-coloring of the product $\omega\times\omega$ that contains no monochromatic copy of $\omega\times\omega$. This raises the question whether a meaningful infinite product Ramsey theorem exists. The Halpern–Läuchli theorem provides such a result for level products of finitely branching trees of height $\omega$ with no leaves. It guarantees the existence of monochromatic strong subtrees (subtrees preserving part of the branching structure of the original trees) of height $\omega$.In this talk I will present a proof of the theorem due to Stevo Todorčević, following a recent exposition by Spencer Unger, which was explained to me by Honza Hubička. If time permits, I will also show how the Halpern–Läuchli theorem implies Milliken's tree theorem, a Ramsey-style theorem in which Halpern–Läuchli plays the role of the pigeonhole principle.