At the state fair, there were $6$ different food items that could be bought. After the receipts were tabulated, it was found that each pair of food items was bought by more than two fifths of the eaters. Moreover, no one bought all $6$ of the food items. Show that at least $2$ people bought $5$ of the food items.

EDIT (2/6): As pointed out below, the problem should originally stated "exactly $2$ people" when it should have said "at least $2$ people".