4. Given the following two propositions about skew lines:
Proposition I: If line $a$ on plane $\alpha$ and line $b$ on plane $\beta$ are skew lines, and line $c$ is the intersection line of $\alpha$ and $\beta$, then $c$ intersects at most one of $a$ and $b$;
Proposition II: There do not exist infinitely many lines such that any two of them are skew lines. Then
A. Proposition I is correct, Proposition II is incorrect
B. Proposition II is correct, Proposition I is incorrect
C. Both propositions are correct
D. Both propositions are incorrect
For proposition $\mathrm{I}$, lines $a$ and $b$ can intersect line $c$ at two different points, so proposition $\mathrm{I}$ is incorrect; for proposition II, construct infinitely many parallel planes, each with a line, where the direction of each pair of lines is different, then any two of them must be skew lines, so proposition II is also incorrect. Therefore, the answer is $D$.