6. Polina has a deck of 36 cards (9 cards of each of the four suits). She selects half of the cards for Vasily and keeps the other half for herself. Next, they take turns placing the cards on the table, one at a time, face up, with Polina starting first. If Vasily can play a card of the same suit or the same rank as the card just played, Vasily scores 1 point. How many points can Vasily guarantee to score at most?
6. 15.

If Polina keeps all the hearts and all the A, K, and Q for herself, Vasily will not be able to score when Polina plays the A, K, and Q of hearts. Therefore, Vasily cannot guarantee more than 15 points.

Next, we prove that no matter how Polina keeps the cards for herself, Vasily can ensure that he scores at least 15 points.

Imagine placing all the cards in a $4 \times 9$ grid (cards in the same column have the same value, and cards in the same row have the same suit).

We will prove that if Polina marks 18 cards as black (the remaining 18 are white), Vasily can always separate at least 15 "good pairs": each pair consists of one black and one white card, either in the same column or in the same row. Therefore, when Polina plays the black card of a pair, Vasily can respond with the white card of the pair, thus scoring.

We classify the columns based on the number of black cards they contain. For example, if a column has 3 black cards, it is called a 3-type column.

Vasily first observes the 2-type columns. As long as there are such columns, he can divide each of them into two good pairs.
Next, he observes the "column pairs" of 0-type and 4-type. Clearly, each such column pair can be divided into 4 good pairs.
He then observes the column pairs of 1-type and 3-type. Each such column pair can similarly be divided into 4 good pairs, as shown in Figure 4.

If none of the above column pairs are left, then by symmetry, we can assume that only 1-type and 4-type "single" columns remain. If there are $a$ 4-type columns and $b$ 1-type columns, then
$$
4 a + b = 3 b \Rightarrow b = 2 a.
$$

In each "column group" consisting of one 4-type column and two 1-type columns, Vasily can ensure at least five good pairs, as shown in Figure 5.
Since
$$
3 a = a + b \leq 9,
$$

there are at most three "bad pairs" left in the grid, meaning Vasily can lose at most 3 points.