Sorry for the delay, my finals weeks have been really hectic, but I just turned in my thesis after having stayed up for 48 hours and ate about maybe 500 calories total today
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I don't know where you're coming from so I don't know what you consider expensive, but my sis and our situation was a little different cuz we were looking for only a 6 month lease and had absolutely no furniture.
Okay, hmm...let's see. there usually are several apartments that are around Little Italy and behind UH hospitals (this is the Fairview / Murray Hill Rd area). One bedroom apartments around there cost about $550-$600 for an unfurnished apartment, included: water and heat (sometimes hot water tank). Usually you have to pay phone, gas, electricity. No A/C or DSL, some have a cable jack, some don't. Not the greatest, but if you're staying for awhile and are willing to fix it up, you could make it homey. Are a bit dirty, but liveable. Others on Euclid Heights or Waldorf towers are also rather expensive like $700-$800 per month.
The ones in a tower complex that have DSL and are more luxurious cost around $700-800 per month, but some if these (Fairhill towers) are a dive. Sometimes there are sublets in houses (or like 4-5 students share a house) that are awesome and go for around $300 per room, I'm thinking of the ones on South Overlook and near Cedar road. Good, clean secure area as well.
You could also look around Coventry, but those are ..eh...
Pretty much if you're looking for something good near campus, stick to south campus. If you start exploring off north campus, it gets shady/ghetto. Also, the med school is closer to south side.
I'd recommend starting to look around now, or late June so that gives you like 4-5 weeks before the med school starts, cuz once August hits, these places will be gone.