insight - Computational Complexity - # Intersection patterns in spaces with a forbidden homological minor

Core Concepts

The paper studies generalizations of classical results on intersection patterns of set systems in Rd, such as the fractional Helly theorem or the (p, q)-theorem, in the setting of arbitrary triangulable spaces with a forbidden homological minor.

Abstract

The paper introduces the notion of a (K, b)-free cover, where K is a forbidden homological minor of the ambient space U and the reduced Betti numbers of the intersections are bounded by b. The main results are:
The fractional Helly number of a (K, b)-free cover is at most µ(K) + 1, where µ(K) is the maximum sum of the dimensions of two disjoint faces in K.
The assertion of the (p, q)-theorem holds for every p ≥ q > µ(K) and every (K, b)-free cover F. This improves upon previous results that required p ≥ q ≥ m, where m is a very large number obtained by successive iterations of Ramsey's theorem.
The proofs use Ramsey-type arguments combined with the notion of stair convexity to construct (forbidden) homological minors in cubical complexes. The paper also discusses connections to topological Helly theorems, stepping-up in combinatorial convexity, and homological VC-dimension.

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Deeper Inquiries

The results presented in the context can potentially be extended to arbitrary triangulable spaces beyond simplicial complexes. Triangulable spaces are topological spaces that can be partitioned into simplices, allowing for the application of techniques and results from simplicial complexes. By adapting the concepts and methods used in the context to the broader category of triangulable spaces, similar results and theorems could potentially be established. This extension would involve generalizing the definitions and properties related to homological minors, colorful covers, and intersection patterns to suit the more general setting of arbitrary triangulable spaces. The key lies in ensuring that the fundamental principles and structures of simplicial complexes can be translated effectively to this broader context.

The improved (p, q)-theorem for (K, b)-free covers has significant algorithmic implications, particularly in the realm of property testing and optimization under constraints. The (p, q)-theorem provides a framework for determining the minimum number of points or elements required to intersect a given family of sets with specific intersection patterns. By extending the range of parameters for which the (p, q)-theorem holds, the algorithmic implications include more efficient property testing algorithms and optimization strategies. The reduced piercing number resulting from the improved theorem allows for more streamlined and effective algorithms in various computational tasks, such as integer programming, discrete geometry, and combinatorial optimization.

The observed gap in the (p, q)-theorem for (K, b)-free covers and the properties of homological minors may indicate a deeper connection between the two concepts. Homological minors play a crucial role in determining the intersection patterns and the fractional Helly numbers of set systems in various spaces. The gap in the (p, q)-theorem, where the range of parameters for which the theorem holds is independent of b, suggests that the structure and properties of homological minors have a significant impact on the intersection patterns and combinatorial properties of the space. Further exploration and analysis of this relationship could potentially lead to a better understanding of the interplay between homological minors and intersection patterns in spaces with forbidden homological minors.

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