USEMO 2020 Problem 2

Calvin and Hobbes play a game. First, Hobbes picks a family $F$ of subsets of $\{1, 2, . . . , 2020\}$ , known to both players. Then, Calvin and Hobbes take turns choosing a number from $\{1, 2, . . . , 2020\}$ which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has $1010$ numbers). Calvin wins if he has chosen all the elements of some member of $F$ , otherwise Hobbes wins. What is the largest possible size of a family $F$ that Hobbes could pick while still having a winning strategy?