A new $k$-partite graph $k$-clique iterator and the optimal colored Tverberg problem for ten colored points

TL;DR

This paper introduces a novel algorithm for verifying the optimal colored Tverberg problem for ten points in the plane, translating the geometric problem into graph theory and efficiently checking for specific clique structures.

Contribution

The paper presents a new algorithm for verifying the absence of k-cliques in k-partite graphs, applied to solve a specific case of the colored Tverberg problem.

Findings

Verified the optimal colored Tverberg partition for 10 points in the plane.

Developed a new k-partite graph k-clique verification algorithm.

Established the intersection property for convex hulls in the specific case.

Abstract

We provide an algorithm that verifies the optimal colored Tverberg problem for $10$ points in the plane: Every $10$ points in the plane in color classes of size at most $3$ can be partitioned in $4$ rainbow pieces such that their convex hulls intersect in a common point. This is achieved by translating the problem to $k$-partite graphs and using a new algorithm to verify that those graphs do not have a $k$-clique.