Kernels of Large-Part Families, Small-Part Ideals, and Cofinality Spectra

This version is a substantial reformulation of the original draft. The original paper was organized around partition-dominating sets (earlier “half-dense sets”), defined by pointwise membership in largest parts together with the bookkeeping condition |𝒫| = n·|H|, and it emphasized existence constructions, dense and comeager instances, forcing preservation, and observations about sphere paradoxes. The present version replaces that framework with the set-theoretically sharper notion of a large-part family with unique maximum parts and its associated kernel Ker(𝒫) = ⋂π∈𝒫 L_π. This removes the tie-case weakness in the original definition and recasts the subject in terms of the small-part ideal generated by the complements S∖L_π. The main theorems are correspondingly different. The present paper proves a presentation theorem reducing kernels to unions of <μ-sized sets, a rigged existence theorem producing families with kernel of full ambient size μ at every infinite cardinal, and a sharp cofinality threshold theorem: if |𝒫| < cf(μ), then every large-part family on a set of size μ has kernel of size μ, whereas for singular μ the full spectrum 0 ≤ ν ≤ μ is realized at every family size from cf(μ) up to μ, even with unboundedly many small parts. In this form the theory is governed by the covering problem for [μ]<μ, and the paper identifies the resulting invariant cov_ker(μ) = cov([μ]<μ) = cf(μ). The revised version also adds an ideal-theoretic and order-theoretic analysis absent from the original draft. It characterizes exactly which ideals arise as small-part ideals, places them canonically below [S]<μ in the Katětov order, and gives a Tukey bound for the associated filter generated by the large parts. It then introduces almost-kernels and shows that on ω and on regular uncountable κ these recover the standard pseudointersection and tower invariants: 𝔭_ker = 𝔭, 𝔱_ker = 𝔱, and 𝔭^ker_κ = 𝔭(κ), 𝔱^ker_κ = 𝔱(κ). Finally, the scope of the paper has been narrowed and sharpened. The forcing section and the sphere-paradox material from the original version have been removed, and the topological part has been reduced to the precise statement that for countable large-part families on Polish spaces, the kernel inherits the corresponding category or measure largeness when the large parts are comeager or conull. The result is no longer a broad narrative about selection phenomena; it is a set-theoretic study of kernels of large-part families, their small-part ideals, and the cofinality spectrum controlling their sizes.