The productivity of the κ-chain condition, where κ is a regular, uncountable cardinal, has been the focus of a great deal of set-theoretic research. In the 1970’s, consistent examples of kappa−cc posets whose squares are not κ−cc were constructed by Laver, Galvin, Roitman and Fleissner. Later, ZFC examples were constructed by Todorcevic, Shelah, and others. The most difficult case, that in which κ=ℵ2, was resolved by Shelah in 1997. In the first part of this talk, we shall present analogous results regarding the infinite productivity of chain conditions stronger than κ−cc. In particular, for any successor cardinal κ, we produce a ZFC example of a poset with precaliber κ whose ωth power is not κ−cc. To do so, we introduce and study the principle U(κ,μ,θ,χ) asserting the existence of a coloring c:[κ]2→θ satisfying a strong unboundedness condition. In the second part of this talk, we shall introduce and study a new cardinal invariant χ(κ) for a regular uncountable cardinal κ . For inaccessible κ, χ(κ) may be seen as a measure of how far away κ is from being weakly compact. We shall prove that if χ(κ)>1, then χ(κ)=max(Cspec(κ)), where: (1) Cspec(κ) := {χ(C⃗ )∣C⃗ is a sequence over κ} ∖ω, and (2) χ(C⃗ ) is the least cardinal χ≤κ such that there exist Δ∈[κ]κ and b : κ→[κ]χ with Δ∩α⊆∪β∈b(α)Cβ for every α<κ. We shall also prove that if χ(κ)=1, then κ is greatly Mahlo, prove the consistency (modulo the existence of a supercompact) of χ(ℵω+1)=ℵ0, and carry a systematic study of the effect of square principles on the C-sequence spectrum. In the last part of this talk, we shall unveil an unexpected connection between the two principles discussed in the previous parts, proving that, for infinite regular cardinals θ<κ,θ∈Cspec(κ) if there is a closed witness to U(κ,κ,θ,θ). This is joint work with Chris Lambie-Hanson.