When former vice president Dan Quayle had a clot in his lungs, he was given a medication called warfarin, brand name Coumadin, to prevent further clots from forming; and was told to avoid broccoli. When Quayle was vice president, (the first) George Bush was president and he hated broccoli, so the recommendation made the front page of the newspapers.
The explanation for avoiding broccoli when you take clot-preventing medicines is not supported by solid science. Warfarin prevents clots by blocking vitamin K. Leafy green vegetables, such as broccoli, contain significant amounts of vitamin K. So, taking large amounts of leafy green vegetables that contain vitamin K could prevent warfarin from doing its job of blocking vitamin K.
However, this is a theoretical concern that occurs rarely, if at all, in patients taking anticlotting medications. Furthermore, everyone who is given warfarin to prevent clots is monitored with daily blood tests at first, and frequently afterward, to see if the warfarin is doing its job of preventing clotting. If large doses of vitamin K in leafy green vegetables prevent the anticlotting effects of warfarin, a blood test should pick it up and the dose of warfarin should be raised. This is just another example of a theoretical concern about a drug that, if followed to its extreme, could impair the health of a person recovering from a heart attack or clot. Leafy green vegetables are loaded with phytochemicals that have been repeatedly shown to help prevent heart attacks and cancers.