Systems of curves on non-orientable surfaces

We show that the order of the cardinality of maximal complete $1$-systems of loops on non-orientable surfaces is ~ |χ|². In particular, we determine the exact cardinality of maximal complete $1$-systems of loops on punctured projective planes. To prove these results, we show that the cardinality of maximal systems of arcs pairwise-intersecting at most once on a non-orientable surface is 2|χ|(|χ|+1).