Thanks for reaching out to us regarding your issues with Red Stem Filaree.  Unfortunately, I did not find any good recommendations for a ground cover to help eradicate this weed.  Since it is an annual, you are likely to have the seed embedded in the ground which could mean you will need to utilize some measures for a number of years.  I am going to attach links to two different fact sheets that will hopefully give you enough information to make the best decision for you.  There are both chemical and organic solutions.  On the organic side, I don't know how large of a space you are dealing with, but pulling, tilling and mulching are known to be pretty effective.  There are several suggestions for chemical approaches, I can't recommend a particular brand of chemical solution, but the fact sheets name ingredients you can look for or ask for at your local garden store.  

Thank you so much!  Is there a benefit to letting it take over?  What are the negatives of it letting it grow?  Since it is all over the area, I probably can’t stop the seeds from coming in my yard even if I want to.  



From what I am reading, red stem filaree is considered a class C noxious weed here in Colorado.