Best way to connect to a really small berg connector
Those are 50 mil spacing double-row pin headers (10x2, 0.05 inch / 1.27 mm pitch). 

Suitable pin strips should be found by searching on your preferred electronic components site.

I found these on eBay "10 PCS 100 Pin 1.27MM Gold Plated Double Row Straight Male Pin Header" for $4.99 including shipping:



The term "Berg" is not applicable, as Berg connectors are a specific brand of pin headers and connectors made by a specific company and have a 2.54 mm (0.1 inch) pitch.