How much water does the grass get in that area when summer hits? Does the rest of your lawn stay green?

Hi. I have an underground sprinkler system so that particular zone normally runs once per week (3/4" of water). If we have a dry, hot spell where temperatures are above 90 for several days, I usually run the zone twice a week.

The rest of the lawn is generally green. There are occasional hints of red thread in other areas thanks to a fairly wet spring but nothing bad enough to justify a fungicide.

Can you post a picture of the area and the tree?

Hi. Sorry for the late reply.

The area in the bottom photo extends to maybe 25 feet from the base of the nearest tree. This entire area was lush with new grass as late as early May. In the bottom photo, the mossy-looking patch near the right edge is actually a stepping stone that borders the lawn area.

Edit: So in the bottom photo, two of the ash trees are visible.
There are are 4 more large ash trees off the top-right of the image and
perhaps a dozen more in my neighbors' yard behind those. To the top of
the photo, near the fence line, are a redbud and a dogwood, neither of
which cast any significant shade directly on the lawn but they do block
low-angle light in those directions.

From what I've read about cool season grasses (Bermuda guy here) most spring seedings fail once the summer heat hits. Most people plant late summer/ early fall once the temps start coming back down, that way the plant has more time to lay down some roots before it gets stressed by the summer heat.

I would consider that heavy shade, not medium...

I'm going to direct you towards NTEP tests performed in dense shade at the Carbondale, IL location. This will be your best bet to find a cultivar that will perform decent in those conditions:

2007-2011 Ful Report page 32 is the one you're interested in

3rd Millenium is available here: http://www.seedsuperstore.com/catalog/p-100012/3rd-millennium-tall-fescue . Looks like it also has good brown patch resistance which may be a concern on damp hot environments.