Matching random colored points with rectangles (Corrigendum)
This paper studies the problem of matching $n$ points sampled uniformly at random in the unit square and colored red or blue, using pairwise disjoint axis-aligned rectangles to match points of the same color, with the goal of maximizing the number $M(n)$ of covered points. Prior work erroneously modeled the matching process as a Markov chain, overlooking its inherent non-Markovian nature. We correct this by formulating it as a first-order homogeneous stochastic process and integrate tools from stochastic geometry, probabilistic methods, and combinatorial matching theory. Rigorously, we prove that for sufficiently large $n$, there exists, with high probability, a monochromatic rectangle matching covering at least $0.83n$ points—i.e., $M(n) geq 0.83n$. This result rectifies the fundamental modeling flaw in prior approaches and establishes, for the first time, an asymptotic lower bound for this problem, thereby significantly advancing the theoretical understanding of random geometric matching.