What kind of ties should interns where

Uh I'll give my two cents on this. 
Firstly, don't buy some cheap satin polyester ties. They dont tie knots well nor do they look good. Satin is a very shiny finish and polyester is a very shiny fabric. Just go 100 percent silk. Don't do wool. That tends to be very heavy in the heat. Trust me. Shatung Silk is a little bit too casual for the office, so just make sure its repp silk or something like that. Grenadine silk is neat, but maybe get a few grey hairs on your head before you pull that out. 

Patterns, I mean you can do small paisley, regimental stripes, repp stripes if you happen to be part of the Brooks Brothers crowd. 

As far as colors go, I mean Dark Red/Burgundy, Dark Blue, Dark Green, and Brown. These are all fine. Again with patterns or stripes you'll have several colors, but as long as nothing is too ostentatiuous you'll be fine.

As far as your last question goes, a Dark blue tie with a suble pattern(paisley or geometric) can work well with it if you want to be monochroamtic, but You can also go with a dark burgundy tie as well. 

Make sure you can tie a decent four-in-hand or double four in hand or half windsor well. Put a dimple in it if you really want. 
Good luck.