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By M3, R2, and K3, one of Allan and Mike is flying to Houston with a black bag; by A3 and K3, the other one of Allan and Mike came from Los Angeles. So by I3, the only person who could be flying to San Francisco is Katie, which means whoever is flying from Los Angeles is flying to Phoenix.

• ASSUME it's Mike who's flying from Los Angeles to Phoenix, and Allan who's flying to Houston with a black bag. Then we have

  Allan flying to Houston with a black bag;
  Mike flying from Los Angeles to Phoenix;
  Rebecca flying from Denver to Charlotte from concourse E;
  Katie flying to San Francisco with a brown bag.

  By J2, Allan isn't flying from Chicago, so Katie is; by elimination, Allan is flying from New York. By J2 and A2, Allan is flying from concourse B. By M2, Mike isn't flying from concourse F, so Katie is. Contradiction with A1.

So it must be Allan who's flying from Los Angeles to Phoenix, and Mike who's flying to Houston with a black bag. By K4 and M4, it must be Rebecca who's flying with a blue bag with Southwest. So

Rebecca is flying from Denver to Charlotte with a blue bag, with Southwest, from concourse E.

By A1 and A2, Katie isn't flying out of concourse B or F, so she must be D. By A4 and K2, it must be Allan who's flying with American Airlines. By M2 and J2, we can now finish off:

Mike is flying from Chicago to Houston with a black bag, with United, from concourse B.
Allan is flying from Los Angeles to Phoenix with a white bag, with American Airlines, from concourse F.
Katie is flying from New York to San Francisco with a brown bag, with Frontier, from concourse D.

By J3 and I3 (for origin, destination, and airline); M4 and J2 (for concourse); and M3, K3, K4, and K1 (for bag colour),

Jeff is flying from Orlando to Miami with a red bag, with Delta, from concourse A.

Partial answer (work in progress)

Denote Mary, Jeff, Allan, Mike, Rebecca, Katie by the letters M, J, A, I, R, K, so that we can denote each of the thirty given statements by a letter-number pair.

First, let's try to pinpoint which of the statements are false, thus reducing this from a liars+logic-grid puzzle to a more standard logic-grid puzzle.

• M1 and K3 contradict each other, so at least one of these must be false.
• I1 and K4 contradict each other, so at least one of these must be false.
• Thus, since at least one of K3 and K4 is true, at least one of M1 and I1 must be false.
• M3 and I1 contradict each other, so at least one of these must be false.

If I1 were true, then M1 and M3 would both have to be false; contradiction, so

I1 is false and I2, I3, I4, I5 are true.

• R1 contradicts the combination of J3 and J5, so either R1 is false or one of J3 and J5 is.
• Since I3 is true, J3 and K5 contradict each other, so at least one of these is false.
• Since I2 is true, M4 and J5 contradict each other, so at least one of these is false.

If J3 and J5 are both true, then R1, K5, and M4 are all false, so K3 and M1 are both true, contradiction. So

one of J3 and J5 is false, which means J1, J2, J4 are all true.

If M1 and M4 are both true, then K3 and J5 are false, so J3 and K5 are both true, contradiction. So

one of M1 and M4 is false, which means M2, M3, M5 are all true.

If K3 and K5 are both true, then M1 and J3 are false, so M4 and J5 are both true, contradiction. So

one of K3 and K5 is false, which means K1, K2, K4 are all true.

We've also now seen that

either M1, J5, K5 are true and M4, J3, K3 are false or vice versa.

• J5 contradicts the combination of R1 and R5, so either J5 is false or one of R1 and R5 is.

Finding the false statements

Denote Mary, Jeff, Allan, Mike, Rebecca, Katie by the letters M, J, A, I, R, K, so that we can denote each of the thirty given statements by a letter-number pair.

First, let's try to pinpoint which of the statements are false, thus reducing this from a liars+logic-grid puzzle to a more standard logic-grid puzzle.

• M1 and K3 contradict each other, so at least one of these must be false.
• I1 and K4 contradict each other, so at least one of these must be false.
• Thus, since at least one of K3 and K4 is true, at least one of M1 and I1 must be false.
• M3 and I1 contradict each other, so at least one of these must be false.

If I1 were true, then M1 and M3 would both have to be false; contradiction, so

I1 is false and I2, I3, I4, I5 are true.

• R1 contradicts the combination of J3 and J5, so either R1 is false or one of J3 and J5 is.
• Since I3 is true, J3 and K5 contradict each other, so at least one of these is false.
• Since I2 is true, M4 and J5 contradict each other, so at least one of these is false.

If J3 and J5 are both true, then R1, K5, and M4 are all false, so K3 and M1 are both true, contradiction. So

one of J3 and J5 is false, which means J1, J2, J4 are all true.

If M1 and M4 are both true, then K3 and J5 are false, so J3 and K5 are both true, contradiction. So

one of M1 and M4 is false, which means M2, M3, M5 are all true.

If K3 and K5 are both true, then M1 and J3 are false, so M4 and J5 are both true, contradiction. So

one of K3 and K5 is false, which means K1, K2, K4 are all true.

We've also now seen that either M1, J5, K5 are true and M4, J3, K3 are false or vice versa.

• J5 contradicts the combination of R1 and R5, so either J5 is false or one of R1 and R5 is.

Now let's get clever. If R4 is true, then A4 and J4 would be completely redundant statements. Assuming none of the information in the puzzle is unnecessary, we must have that

R4 is false and R1, R2, R3, R5 are true,

which means

M1, J5, K5 are false and M4, J3, K3 are true.

Now the only false statement we still need to find is Allan's.

Deducing the correspondences

By M5, R5, R1, I3, and I2,

Mary is flying from Dallas to Vegas with a green bag, with JetBlue, from concourse C.

Now we know which of Allan's statements is false:

A5 is false and A1, A2, A3, A4 are true.