A Proof of Paták's kb+1 Conjecture for Constrained Stars, with Improved Bounds for Complete Graphs

In an Oberwolfach report from 2020, Paták considered closure operators on topological spaces whose closures have at most b path-connected components. He noted that C(b+1,2)(k−1)+b+1 points suffice for a constrained drawing of the star K_{1,k}, conjectured that kb+1 points suffice, and asked two further questions: whether the bounds for complete graphs K_n can be improved, and whether the method extends to higher homology or homotopy. In the journal version (J. Graph Theory, 2025) the conjecture is stated for b-iatlon graphs and proved there for b ≤ 2. We prove the conjecture for all k and b. Every b-iatlon graph with kb+1 vertices contains a constrained copy of K_{1,k}. For every closure operator as above, every set of kb+1 points admits a constrained drawing of K_{1,k}. The bound kb+1 is sharp in both settings. The proof rests on a colouring lemma for families of graphs indexed by the subsets of a finite set and growing with the subset; it extends the bound χ ≤ Δ+1. As a consequence, 1+b+⋯+b^{n−1} points force a constrained copy of K_n, improving the previous bound O(b^{2n−3}). Combined with Paták's topological arguments, this lowers his bounds on Radon and Helly numbers in R^d from O(b^{2d+3}) to O(b^{d+2}), and his bound on Radon numbers on a fixed closed surface from O(b^6) to O(b^3). In the b-iatlon setting, Ramsey numbers give lower bounds, which show that the exponent n−1 is optimal for n = 3, 4. Exact values for complete graphs remain open, and the question on higher homology and homotopy is not addressed. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-1703876-006 (Oberwolfach Reports, "Helly numbers of disconnected sets", P. Paták, Report 30/2020, pp. 1500–1501).