2025/12/29 by Alan Dow, Dow, Alan, István Juhász +1
Computer Science · Economics, Econometrics and Finance · Mathematics · #54A25 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #Economic theories and models #FOS: Mathematics #General Topology (math.GN) #Logic (math.LO)
paper · doi:10.48550/arxiv.2512.23544
openalex publication_date 2025/12/29 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28
The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality 2^< \mathfrakc = \mathfrakc with the following property (*) of partitions of pairs of \mathfrakc: \smallskip (*) For any coloring (or partition) k : [\mathfrakc]2 → 2 either there is a homogeneous set of size \mathfrakc in color 0 or there is a set S ∈ [\mathfrakc]^\mathfrakc such that for every countable A ⊂ S there is β∈ \mathfrakc for which A ⊂ β and k(\α, β\) = 1 for all α∈ A. \smallskip (*) plus 2^< \mathfrakc = \mathfrakc together then imply that for every topological space X of countable spread, i.e. not containing any uncountable discrete subset, |X| ≤ \mathfrakc if it is Hausdorff and o(X) = \mathfrak c if it is also infinite and regular. Here o(X) denotes the number of all open subsets of X.