Step 1: Understanding the Question.
We need the name of the concept that describes an ongoing co-evolutionary arms race between a pathogen and its host, where each side keeps changing to gain the upper hand over the other.
Step 2: Key Concept or Approach.
The Red Queen hypothesis takes its name from the Red Queen in Lewis Carroll's Through the Looking Glass, who tells Alice that here you have to run as fast as you can just to stay in the same place. In biology this idea means that a host and its pathogen (or a predator and its prey) are locked in constant evolutionary change, each new defence in the host is met by a new counter-attack in the pathogen, so neither side gains lasting ground even though both keep evolving.
Step 3: Detailed Explanation.
Check the other named ideas in turn.
The Green Beard model describes a gene that produces a visible marker (the metaphorical green beard), lets other carriers of that gene recognise each other, and causes carriers to help each other. It is a model of kin-like recognition and altruism, not of a host-pathogen race.
Sympatric speciation describes new species forming from a single ancestral population without any geographic separation, usually through strong disruptive selection or assortative mating within the same area. It is about how one lineage splits into two, not about an ongoing host-pathogen conflict.
Reinforcement describes an increase in premating reproductive isolation between two populations that live in the same area, caused by selection against low fitness hybrids. Again this is a speciation concept, not a description of host-pathogen coevolution.
None of these three match the arms-race pattern in the question, while the Red Queen hypothesis was built specifically to explain why hosts and pathogens (and predators and prey, or parasites and hosts more broadly) must keep evolving new traits just to hold their ground against each other.
Step 4: Final Answer.
The concept describing a co-evolutionary arms race between a pathogen and its host is the Red Queen hypothesis.
\[ \boxed{\text{Red Queen hypothesis}} \]