I think they've corrected it. Mine came out ok.

I've tried that, doesn't work....don't know why

Although "Ich habe eine sehr gute Verbindung mit ihm" is not a usual phrase in german, it is still correct in this case. Just report it. Btw. you can also say "..gute Verbindung zu ihm" or "Ich habe einen guten Draht zu ihm." But Duolingo is not accepting it yet.

According to dictionary, Verhältnis is grouped with words implying a mathematical ratio or relationship while Verbindung is more a connection or personal relationship. It seems to me the latter would be the preferred word in this case.

Ich habe ein sehr guteS Verhältnis mit ihm because it is "das Verhältnis"