First-kind colorful (p,q) theorem for separated d-intervals

Establish the first-kind colorful (p,q) theorem for the convexity space of usual separated d-intervals, namely (R^d, C_equiv(R^d)), within the framework of existing results for d-intervals.

Background

The paper studies Helly-type properties of the convexity space whose ground set is Rd, viewed as d disjoint levels, and whose convex sets are usual separated d-intervals. It proves a first-kind colorful (p,q) theorem for the more general -convexity space (P, C_equiv(P)) when p geq q geq 2d.

The Discussion notes that the known transversal and colorful transversal bounds for separated d-intervals yield improved bounds for the ordinary (p,q) theorem and the second kind of colorful (p,q) theorem. However, the corresponding first-kind colorful (p,q) theorem remains unresolved when working within the existing d-interval results, making the derivation of such a theorem an explicitly stated open problem.

References

While Theorem~\ref{tar} and Theorem~\ref{fri} yield improved bounds for the $(p,q)$ theorem and the second kind of colorful $(p,q)$ theorem for the space of usual separated $d$-intervals, that is $(Rd,\mathcal C_{\equiv} (Rd))$, obtaining the first kind of colorful $(p,q)$ theorem remains unknown within the framework of existing results for $d$-intervals.

— Helly-type theorems for separated $d$-intervals  (2501.03207 - Rao, 6 Jan 2025) in Discussion