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Weakly compact cardinals: A combinatorial proof

We give here a direct purely combinatorial proof that weak compactness is equivalent to a combinatorial property (2). This property (2) is apparently stronger, and from it, all other usual equivalent definitions and usual properties of weakly compact cardinals can be deduced. So this proof may be useful for books which want to present weakly compact cardinals, but not logic. A direct simple proof of a weaker implication (e.g., weakly compact μ is not the first inaccessible, and every stationary set has an initial segment which is stationary) was given by Kunen [K] and independently by the author [Sh]. Baumgartner [B] had another proof. We were motivated by the manuscript of Erdös, Hajnal, Mate and Rado's book on partition calculus, and by conversations with A. Levi who was writing a book on naive set theory. Notation. Let i, j, α, β, γ, η, σ be ordinals, μ be a cardinal, f, g be functions. Let cf α be the cofinality of α. A partially ordered set T is a tree if for any a ∈ T, {b: b < a, b ∈ T} is well ordered; its order type (an ordinal) is called the level of a, and Tα is the set of a ∈ T of level a.