DNA: Uncovering hidden matches

I am still analyzing and comparing the genetic matches of my mother and her brother at 23andMe.  At this site, you find people related to you in the "Relative Finder."  My mother has over 1300 matches.  Her brother has around 500.  My mother has few Irish matches.  Her brother has lots of Irish matches that also match her, but were previously not known, because she reached her limit of matches.  It is great that I now have access to these Irish cousins and have some good leads to follow.  The site does not produce such matches for my mother because the number of revealed matches is capped at an undisclosed number.  Her slots were taken by matches with multiple segments.  Most of these Irish matches are identical on only one segment; therefore, their percentage of matching was less than her 1300 multiple-segment cousins and they did not make the cut.

This genetic cousin matches both my mother and her brother.  This match is revealed in his account, but not hers.

The first hint I had about missing matches in the system was with my father's third cousin.  I compared my mother to my father's third cousin by chance and was surprised that they matched.  Neither appears in the other's Relative Finder.  As explained above, my mother is maxed out on matches, but this cousin only has about 600 matches, no where near the limit.  Yet because my mother was maxed out in her account, this cousin was prevented from seeing her in his account.  I wonder how many more cousins she has in the database that are not discoverable.

23andMe comparison of known third cousins, matching on chromosomes 13 and 21.
Also matches my mother through unknown connections.

To take this a step further, at 23andMe this third cousin and I share no genetic material.  If you compare us at GEDmatch, we do match, but strangely, not on the large areas on 13 and 21 that my father shares with this third cousin and not on the smaller segment on 18 that he shares with my mother.  He and I share a 5.8 cM match on chromosome 1, which is another area where he matches my mother.  This segment is large enough that 23andMe should have picked it up.  (Their minimum threshold is 5 cM.  Matches smaller than this may be coincidental.)

GEDmatch comparison between R.S. and me.
Of the nine identical segments, two can be identified as from my father and one from my mother.
Note that I did not inherit any part of the three large segments that R.S. shares with my parents.

GEDmatch comparison between R.S. and my uncle.
The two tiny segments may be coincidental and not suggestive of shared ancestry.

GEDmatch comparison between R.S. and my father.
They are third cousins and this analysis is consistent with that documented relationship.
The two larger segments on 13 and 21 are also reported in 23andMe.

GEDmatch comparison between R.S. and my mother.
The largest match on chromosome 18 is the one match identified by 23andMe.