The FBI years
18 episodes of The Mentalist carry this thread, 18h in all — 3 of them are in The Red John arc.
Arc membership is derived from each episode's own title and summary text. It shows where a thread is foregrounded, not every episode it touches.
In airing order

Jane and the CBI narrow down the list of suspects after a disgraced clothing designer on the verge of a comeback becomes a fashion victim. Meanwhile, FBI Agent Darcy continues her investigation into finding Red John.

After another attempt to find Red John fails, Patrick Jane hits rock bottom at a Las Vegas club in the fourth season finale.

While Jane tries to decipher Lorelei's connection to Red John, the CBI team is forced to partner with the FBI to solve the murder of a hotel employee.

A bank employee is murdered during a heist and Jane and the team must work to track down the killer. Meanwhile, Lisbon is pulled deeper into the CBI/FBI rivalry.

The FBI reluctantly enlists Jane's help to find a missing computer programmer, but he refuses to help unless Lisbon is brought on board as well.

The FBI is tasked with finding the killer of several DEA agents, and Jane goes on a date with a beautiful woman associated with the case.

Jane and the FBI investigate the murder of a high-tech cartographer who may have unearthed a spy ring before his death. Meanwhile, Rigsby and Van Pelt find themselves on the trail of an alarming conspiracy.

Jane and Fischer investigate a murder at an oil fracking site, while Rigsby and Van Pelt travel to Austin to help investigate suspects in the string of attacks on former CBI team members.

Jane enlists his colleagues in a dangerous sting operation to catch a deadly team of art thieves. Meanwhile, sparks fly between Lisbon and an agent from the FBI's art squad.

Jane and the FBI take up the cause of a death row inmate who has two days to prove his innocence before his sentence is carried out.

Jane's freedom is at stake when a grand jury is empanelled to decide if he should be tried for the murder of Red John. Meanwhile, Lisbon tries to find out how Jane feels about the possibility of her leaving Austin with Marcus Pike, and Cho and Abbott follow new leads in the human trafficking ring they recently uncovered.

Jane and Lisbon decide to keep their new relationship a secret from their colleagues as they investigate the murder of an undercover agent. Also, young, ambitious agent Michelle Vega joins the team.

Jane and Lisbon travel to Beirut where they must work with former adversary, Erica Flynn, who agrees to help the FBI apprehend her boyfriend, a courier for terrorist organizations, in exchange for amnesty for her past crimes.

The team investigates a murder linked to a counterfeit diamond operation, but illness forces Jane to direct his colleagues from the office as he recuperates. Meanwhile, Abbott is concerned that transgressions in his past could harm his wife's chance to obtain a prestigious job in Washington, D.C..

The FBI is brought in to consult on a botched DEA raid, but when Abbott's former boss tries to blackmail Abbott into keeping the agency's missteps quiet, he is forced to choose between his career and morals.

When Abbott's old boss, Bill Peterson, threatens Abbott's career – and possibly his freedom – Jane devises a risky plan to protect his friend's future.

A young man claiming to have psychic powers offers his help to the FBI in solving a pair of murders, but when he leads them to additional victims, Jane and the team begin to suspect that he is in fact the killer they're hunting.

Jane agrees to once again pretend to be a renowned psychic and be used as bait in an attempt to lure out a serial killer and give the FBI a chance to apprehend him.