Learjet 75 Captain reported taking evasive action due to ATC failure to provide separation resulted in an NMAC.

Narrative

On Day 0; at about XA:22 ZULU; while being vectored with ZZZ Departure on frequency XXX.XX; we were at 6;000ft. We were instructed to turn heading 120 and climb to 8.000 feet. At 8;000 feet; we detected on our traffic screen another aircraft flying towards us with little vertical separation from us. The controller gave us a heading of 280; making us fly to the conflicting aircraft. The TCAS detected the conflicting aircraft and instructed us to descend. The controller instructed us to climb to 16;000 feet; but we were already replying we were on TCAS RA. We stopped the turn; descended to about 7;000 feet and near missed the aircraft to our right. The controller recognized the conflict and gave us a descent (we were already reaching 7000 feet). After clear of conflict we did turn to the last assigned heading of 280 and remained 7000 feet waiting for instructions and some explanation of what just had happened. The controller continued vectoring us as if NOTHING happened. We did not complain to the traffic controller because the frequency was saturated.