Asymptotic Hyper-Ramsey Thresholds for Monochromatic Substructure Emergence in Stochastically Evolving Xeno-Social Networks

Mathematical Applications of Science Fiction The phenomenon of emergent order within chaotic systems remains one of the central pillars of combinatorial theory. Since the foundational work of Ramsey, it has been understood that sufficiently large structures must contain highly ordered substructures—a principle succinctly summarized as “complete disorder is impossible.” However, classical Ramsey theory operates primarily on static objects. In the context of Xeno-Social Networks—conceptual frameworks modeling the interaction topologies of non-terrestrial, high-complexity agent swarms—the underlying graph structure is not static but evolves stochastically over time. This manuscript addresses the Dynamic Hyper-Ramsey Problem: given a k-uniform hypergraph Hn on n vertices, evolving according to a stochastic edge-exposure process Xt, at what critical threshold does the hypergraph guarantee the existence of a monochromatic copy of a target structure F under any r-coloring of its hyperedges? We formalize this using the language of random hypergraphs H(k)(n, p) and extend it through a functorial mapping to a time-dependent category. The primary contribution of this work is the derivation of the Asymptotic Hyper-Ramsey Threshold, denoted pcrit, and the proof that the transition from “Ramsey-avoiding” to “Ramsey-inevitable” is sharp.