OK, so I know this is something of a zombie thread, the post before the last one was almost a year ago.
However, I don't remember seeing this at the time, and since I already have a Gedmatch account (admittedly, not used in about a year or so) I thought I'd have a look.
These are very small segments of DNA indeed. There are various choices you can look at from 10 cM down to 0.5 cM.
In comparison, Ancestry use a minimum figure of 8 cM of matching DNA to show you as a DNA match to another person.
And with shared matches, Ancestry only shows a person as a shared match if you share more than 20 cM of DNA with that third person.
Having said all that, it did show matches for my dad at the 5 cM level. He matched on two segments, each at 5 cM, with an 8,000 year old DNA from Luxembourg and then also from one segment on a 13,700 year old DNA from Switzerland.
At a lower level, 3 and 4 cM, he also had matches with multiple DNA samples from Germany, Hungary and Russia and also one match from Anglo-Saxon Cambridgeshire. The only match with one of the Romans in York comes at the 2 cM level.
In contrast, my mum has a match at the 5 cM level with an Irish DNA from 2,000 years ago. At the lower level of 3 or 4 cM she also has multiple matches from Ireland, Germany, Hungary and Russia but also Georgia and Spain - as well as Luxembourg and Switzerland.
Even at the 2 cM level my mum doesn't have any links to the Romans in York, but does have a match with early Anglo-Saxon DNA (400-525 CE) from Cambridgeshire.
Then, even at 1 cM there is very little (so little that it's likely just random) that links my mum to those Romans so I'm pretty sure she's got no connection.
With my own DNA, I share DNA at the 5 cM level with the Swiss match from my father and also the Irish match from my mother.
Although, just to show you that at this very low level it can be random. When I look at my DNA matches it shows that I share 3 cM with an 8,300 year old DNA match from Anatolia, Turkey.
However, neither of my parents show this same "match". It's not a real match it just happened that I inherited a very small segment from my parents that randomly happened to match an entirely unrelated DNA.