I think the taxpayer can claim the girlfriend as a dependent. Social Security doesn't count towards income limit. But has to be a blood relative to claim Head of Household .
Here's a FAQ for Head of Household
https://ttlc.intuit.com/community/family/help/do-i-qualify-for-head-of-household/00/25539
Who is a qualifying person for Head of Household
https://ttlc.intuit.com/community/family/help/what-is-a-qualifying-person-for-head-of-household/00/2...
