A LJ60 encountered multiple traffic conflicts which required combining TCAS advisories with see and avoid maneuvering to ensure separation.

Narrative

We were assigned 6000 FT and 080 heading; followed by a call from Luke Approach about traffic at our 11 o'clock 1000 FT below us. I was the non-flying pilot and was in the right seat; therefore I could not make visual contact with the Cessna at 11 o'clock below the airplane. A TCAS target was displayed at this location relative to our airplane and showed a hollow blue diamond climbing at 1000 FT below us. While searching for the Cessna visually; I noticed two aircraft at out 1 o'clock; approximately 4 miles away; level with us. I immediately queried Luke Approach as to the 1 o'clock traffic which was crossing from our right-to-left (coming from the direction of Luke AFB) the controller stated there was a flight of F-16s at our 1 o'clock position; so I asked him if they intended to climb. At this point the Pilot Flying (PF) made visual contact with the Cessna at 11 o'clock; noticed it was climbing AND heading towards us; and began a gradual climb to avoid. Then the TCAS RA announced 'Monitor Vertical Speed' (with red from 0 to 1500 FPM climb) I immediately told the pilot flying to descend; but he was continuing to avoid the Cessna below us; so I said much louder: 'Descend now' and 'we have traffic above us; descend'. By this time our aircraft had climbed about 700 FT to avoid the Cessna and the F-16s were climbing also; but at a greater rate than we were. The PF began a descent to avoid the F-16s and we missed the first 2 by about 1/2 mile laterally and 300 FT vertically. Two more F-16s crossed directly overhead about 1000 FT above us. Luke Approach notified us of traffic (Cessna) that was less of a threat than the F-16s; but did not notify us of the F-16s until I queried them. The simultaneous nature of multiple aircraft converging on the same point caused a little confusion and a lot of communications problems. In the end we used 'see and avoid' to safely place the airplane between all the other traffic.