If I'm using two prusik's to climb a rope (one as a foot loop and one attached to harness directly) What's the minimum diameter of cord I should be looking for to remain safe?
EDIT: When the question was posed I misinterpreted it to be asking the diameter of the "Climbing Rope" not "Cord" However my answer for Cord diameter is also listed as "2mm less than cord diameter and not less than 5mm"
I would not go any lower than 7.5mm*, it is a common diameter used by rescue technicians.
There are several reasons why I recommend 7.5mm, listed below.
You need to remember and account for the diameter of your prussik, which following the general rule of thumb, is at least 2mm less than the rope diameter. A 5mm (Static,Tech or Nylon) prussik supplies a good amount of grip. If you chose a rope of smaller diameter your prussik will be approaching some very small diameter cord.
A 7.5mm such as the Sterling Escape Tech provides good abrasion resistance, and has the ability to withstand falls, something a rope of smaller diameter may not be able to provide.
Any smaller than 7.5mm the rope itself will be difficult to ascend, and if you chose to descend on it the rope will have a tendency to coil up and even knot itself.
*According to numbers a 6mm static cord can hold 1600 lbs however I would not recommend using it for the purpose of ascending, let's keep it for hoisting the bear barrels into the trees.
If you attempt to ascend a rope using only friction knots please use a safety back up such as the system detailed below!
As a safety precaution tie off the slack end of the rope, using a figure eight on a bite, at progressive intervals and clip to your harness. To avoid having a clutter of these knots all hanging off your harness you might choose to untie and release each previous one. Note, however, some climbers prefer the clutter to potentially getting the trailing rope snagged far below.
I am going to approach this question differently. Since the question is marked with the rock-climbing tag the use of a 7.5mm Technora escape rope that AM_Hawk's answer focuses on seems unlikely. Also, the method by which he arrives at a minimum Prusik cord size doesn't address certain issues.
All else being equal a smaller cord for a friction knot, such as a Prusik, will grip better than a large one. I have seen several guidelines, such as use cord no larger than 70% of the diameter of the rope being ascended, but truly this is only a guideline. A stiff and slick cord will have much less purchase than a limp and rough one, therefore maximum diameter is determined pragmatically by what will grip. There are cords made specifically for friction hitches, and these typically limp cords will work better even in larger diameters than will stiff accessory cord not intended for the application. For example, BlueWater 8mm Dynamic Prusik Cord is limper and will grip better than their 6mm Accessory Cord.
As far as minimum size, there really is none from a friction standpoint as the smaller cord grips better; it is entirely possible to use a shoelace, paracord, a string from your chalk bag, etc., so long as it can hold your weight. One would be a fool to have these thin cords be his only attachment to the rope of course. You can protect the ascent by progressively clipping in to loop knots in the rope, as AM_Hawk described, or you could use a non-toothed progress-capture device such as a Petzl Shunt. (Please don't implement a system without a full understanding of the devices, failure modes, and risks involved; this is merely an example.)
Perhaps you are seeking a minimum size of friction hitch that is safe as a primary connection point. That is hard to answer. One possible failure mode is burn-through; that is, if the knot slips the friction will melt through the cord long before you deck. Hypothetically a smaller cord, if the loop is appropriately sized and the knot well dressed, might be less susceptible to this as it should be less likely to slip, but a larger cord stands a better chance of arresting the fall before it melts or tears through. There is also the consideration of fusing. Several test reports describe the friction hitch fusing to the main line after slipping; while this is less immediately terminal than burn-through it could still leave you stranded.
I was unable to find dynamic (drop) testing of small diameter Prusik cords. However, here are some resources that seem relevant.
A test report for twin 8mm Prusiks at rescue loads: CMRU July 1997 Drop Tests.
An extensive test report for using a Purcell Prusik as a personal anchor, including 6mm cord: Rigging for Rescue -- Lanyards Part II: An Examination of Purcell Prusiks as Personal Restraint Lanyards. (The circumstance is almost entirely different from rope ascension: the fall factors are much higher but slip is inherently limited.)
Drop testing on arborist's equipment: Ropes and Friction Hitches used in Tree Climbing Operations -- Paolo Bavaresco, Treevolution