There's a fan theory that Petyr Baelish is the secret father of Robert Arryn, the son of Lysa Tully and presumably Jon Arryn.
He warmly greets Robyn each time he sees him, and this is something he does not do for others. He's not affectionate with anyone else, but he hugs this young boy.
He once impregnated Lysa Tully, but forced her to drink an abortifacient poison to cause a miscarriage. (Which also removes Jon Arryn's doubts about whether his wife was fertile enough to get pregnant.)
Jon Arryn was nearly 70 when he died, and Lysa said he was too old to get her pregnant.
Baelish chose the mockingbird as his sigil, and the mockingbird is known for getting other birds to raise their eggs.
Robin's hair color matches that of Petyr Baelish, but does not match that of Jon Arryn. Jon Arryn was blonde as a young man.
So did Baelish ever hint or say that sweet young Robin was his secret son?