Medium Large Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP)
Situational Awareness
ATC Issue All Types; Conflict Airborne Conflict
Flight Crew Took Evasive Action
Human Factors; Procedure
Inbound to KOA at FL190; we were given PD to 13000 and were told [another aircraft] was westbound out of KOA. Through 14000 were given a vector of 090 and PD to 11000. [The other aircraft] then called us in sight and they were told to maintain visual with us. HCC then gave us PD to 3000. I told my co-pilot we were going to stay at 11000 until we had [the other aircraft] in sight or once they had passed.  At approximately the same time we got a TCAS (TA) telling us traffic was at one o'clock; 500 feet below us and 5 miles. We could not see the traffic because of scattered clouds. Shortly thereafter we received another TCAS (RA) 'Monitor vertrical speed' telling us not to descend; traffic was now at two miles. [The other aircraft] passed 500 feet directly beneath us. Once clear of conflict was announced we started our descent to a normal landing in KOA.Not sure why HCC vectored us toward traffic and then gave us a clearance to descend through their altitude.
[We] departed KOA this afternoon [and] contacted HCF center receiving a transponder code. A few minutes later we were instructed to cross the PHKO localizer and remain west of the 351-R off KOA. When we were 12 miles west of the 351-R HCF center called out [air carrier traffic] at 11 O'clock. I reported; 'traffic in sight at 11 O'clock but unable to confirm [type]'. We were instructed to maintain VFR at or below 10500; which we did. [The other flight] reported an RA to center. Center responded with; 'Roger'. We started a descent a few seconds later; they flew right over us with a separation of 500 vertically.
Captain of a medium transport aircraft and Captain of a small transport reported coming within 500 feet of one another in the vicinity of KOA.