I would expect them to claim title to the disputed area.Quoting blueeagle
To be clear, does their deed state that they own the disputed area? If not, how did their surveyor come up with boundary lines for their lot that includes the disputed area? If so, what did you find in the county records as to when and how the discrepancy arose. For example, if we're talking about a discrepancy that occurred when the two parcels were split from the same lot, the neighbor could have as much support for his claim to the land as you have -- or more, given that he can point to forty+ years of the use of the land by himself and his predecessors in title.