Professor Doublebrain was severely ill last week, and he had to spend the six days from Monday till Saturday in the hospital. Luckily he has fully recovered by now. He told us that on each of these six days of illness he was visited by some of his 20 closest friends; and of course many of these friends visited him more than once. Professor Erasmus thought about this for some time, and then told me:
"No matter how and when and how often these 20 friends were visiting Professor Doublebrain on these six days of illness, one will be able to pick two days $D_1$ and $D_2$ and pick five friends $F_1,F_2,F_3,F_4,F_5$, such that the following holds: either all five friends visited Doublebrain on both days $D_1$ and $D_2$, or all five friends visited Doublebrain neither on day $D_1$ nor on day $D_2$."
Is this statement correct, or has professor Erasmus once again made a mathematical blunder?