FO OF AN EMB120 MADE A STEEP EVASIVE ACTION TURN TO AVOID AN EXECUTIVE JET APCHING FROM THE OPPOSITE DIRECTION FOR A DOWNWIND LEG TO THE RWY ON WHICH HE HAD JUST DEPARTED. TWR CTLR HAD NOT ADVISED OF THE APCHING JET.

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OUR DEP OFF RWY 19 AT MEI RESULTED IN A TA AND A NEAR MISS DUE TO OUR PROX TO A BUSINESS JET ON APCH TO RWY 1; AND A PA28 THAT HAD JUST DEPARTED RWY 22. DURING TAXI OUT; MEI TWR CLRED THE PA28 FOR TKOF FROM RWY 22. APPROX 2 MINS LATER WE WERE ISSUED DEP INSTRUCTIONS 'CLB MAINTAIN 5000 FT; R TURN TO 270 DEGS; CLR FOR TKOF RWY 19.' THE TWR DID NOT PROVIDE ANY ADVISORIES OF THE INBOUND BUSINESS JET ON THE PA28'S DEP INSTRUCTIONS. ONLY OUR SITUATIONAL AWARENESS PROVIDED US WITH THE KNOWLEDGE THAT THE PA28 WOULD BE SW OF THE ARPT. THE CAPT; AS PNF; SELECTED THE TCASII 5 MI RING; WHICH SHOWED THE PA28 AT 1-2 O'CLOCK POS; 1-2 MI; AND ABOVE US; CLBING. SHORTLY AFTER TKOF THE TWR DID ADVISE US OF THE PA28. WE WERE UNABLE TO MAKE VISUAL CONTACT AND; THEREFORE; DID NOT IMMEDIATELY TURN TO 270 DEGS; IN AN EFFORT TO MAINTAIN SAFE SEPARATION. PRIOR TO THE HDOF TO DEP CTL; WE WERE UNABLE TO MAKE VISUAL CONTACT WITH THE PA28 AND NO MENTION OF THE JET WAS EVER MADE. DUE TO OUR EFFORT TO ACQUIRE THE PA28 VISUALLY; WE DID NOT SEE THE JET ON THE TCASII. AS WE TOGGLED UP MEI DEP; THE CTLR WAS ALREADY CALLING US AND ASKED IF WE HAD 'INITIATED OUR TURN TO 270 DEGS?' AT THAT TIME ;I; AS PF; NOTICED A SECOND TARGET ON THE TCASII AND IMMEDIATELY BEGAN A STEEP EVASIVE TURN TO 270 DEGS. SIMULTANEOUSLY; THE TCASII PROVIDED THE TFC ALERT AND THE CTLR ADVISED 'TFC 12 O'CLOCK POS; YOUR ALT; LESS THAN 1 MI; OPPOSITE DIRECTION.' LESS THAN 5 SECONDS LATER; THE CAPT DID SEE THE JET PASS BEHIND US AND TO OUR L; APPROX 200 FT BELOW OUR ALT. WE THEN BEGAN TO SEARCH AGAIN FOR THE PA28. DEP INSTRUCTED US TO MAINTAIN 3000 FT. WE DID AS INSTRUCTED AND DID ACQUIRE THE PA28 AT 11 O'CLOCK POS; LESS THAN 1 MI; AND ABOVE OUR ALT. WE WERE ABLE TO PASS BEHIND THE PA28 WITHOUT HAVING TO CHANGE OUR 270 DEG HDG. THE CAPT THEN CONTACTED DEP AND EXPLAINED THAT WE DELAYED OUR TURN MOMENTARILY TO TRY TO ACQUIRE THE PA28 VISUALLY AND THAT WE HAD NO KNOWLEDGE OF THE JET ON APCH FOR RWY 1. THE CTLR REPLIED 'I DON'T BLAME YOU ONE BIT;' AND HE APOLOGIZED; AND SAID THAT HE WAS 'GOING TO GIVE MEI TWR A CALL RIGHT NOW AND HAVE A TALK WITH THEM ABOUT THAT.