Compact metric spaces with infinite cop number
Agelos Georgakopoulos

TL;DR
This paper disproves Mohar's conjectures by constructing a compact geodesic metric space, specifically on , with an infinite cop number, challenging previous assumptions about pursuit-evasion games in such spaces.
Contribution
It provides a counterexample to Mohar's conjectures, showing that finitely many cops may not always suffice in compact geodesic metric spaces.
Findings
Constructed a metric on with infinite cop number
Disproved the conjecture that finitely many cops can always win
Raised new questions about pursuit-evasion in metric spaces
Abstract
Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on with infinite cop number. More problems are raised than settled.
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Taxonomy
TopicsArtificial Intelligence in Games · Gambling Behavior and Treatments · Mathematical Dynamics and Fractals
