Filter Django database for field containing any value in an array

Make successive calls to filter, like so:

queryset = Profile.objects.all()
strings = ['keith', 's', 'thompson']
for string in strings:
    queryset = queryset.filter(full_name__icontains=string)

Alternatively you can & together a bunch of Q objects:

condition = Q(full_name__icontains=s[0])
for string in strings[1:]:
    condition &= Q(full_name__icontains=string)
queryset = Profile.objects.filter(condition) 

A more cryptic way of writing this, avoiding the explicit loop:

import operator
# ...
condition = reduce(operator.and_, [Q(full_name__icontains=s) for s in strings])
queryset = Profile.objects.filter(condition)