B737 Next Generation Undifferentiated
Flight Crew Multiengine; Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument
Flight Crew Total 12000
Situational Awareness; Time Pressure; Workload; Communication Breakdown
ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Fuel Issue
Flight Crew Diverted
Procedure; Human Factors; Weather
Declared minimum fuel with Approach after rolling EFC (Expect Further Clearance) given during holding in Center ZZZ airspace. We were trapped into assuming our flight would be next for the ZZZZZ arrival into ZZZ while holding over ZZZZZ1 fix. We did not divert to our alternate; ZZZ1; due to ATC informing us that ZZZ was accepting aircraft and that we would be beginning the STAR soon.  We kept getting 10-20 minute rolling EFC's and when I advised Center we need to begin the STAR soon; we were told you are next for the airport. This kept us from diverting because of our belief that we would start the STAR at any minute.This low fuel event occurred in large part due to Center Y not allowing aircraft into their sector. Center X advised numerous aircraft; including our flight; that Center Y is being 'very' uncooperative about allowing aircraft into their sector and holding closer to ZZZ. Also; there was not enough information about when our actual time for the arrival would begin. Also too many false starts to begin the STAR was issued putting our flight into a position where diverting was the same fuel burn as arriving at destination. Added to this was the fact that Approach also vectored us off the STAR for spacing.It is ultimately the responsibility of the Captain to make the decision to divert early enough to prevent falling into this trap from ATC. ATC does not ultimately know or care about aircraft fuel state. Never let ATC lull the flight crew into thinking that 'you are next for the arrival' and thinking that no other factors or events might force them to remind that promise.
B737 pilot diverted due to low fuel situation.