Pigeonhole principle: Show that out of 7 lines in a plane we can choose 3 paralel lines or 4 concurrent lines.

Question

There are $7$ lines in a plane. Show that we can always pick either $3$ paralel lines or $4$ concurrent lines. Show that it won't work for $6$ lines.

I have no idea where to start with this.

Answer 1

I believe you meant to prove that there are either $3$ parallel lines or $4$ lines pairwise intersecting. Let’s divide all the $7$ lines into groups of parallel lines. Let there be not $3$ parallel lines. Then each group contains at most $2$ lines. So since there are $7$ lines in total there are at least $4$ groups (by pigeonhole principle). Now take a line from each of $4$ groups and you get $4$ lines pairwise intersecting