19. (15 points) In a certain grade, $n$ students participate in Chinese and

Mathematics exams, with scores ranging from 0 to 100 for each subject. Suppose no two students have exactly the same scores (i.e., at least one subject score is different). Additionally, “A is better than B” means that student A’s scores in both Chinese and Mathematics are higher than student B’s scores in both subjects. Question: What is the smallest value of $n$ such that there must exist three students (let’s call them A, B, and C) where A is better than B, and B is better than C.
19. Establish a Cartesian coordinate system $x O y$.

If a student's Chinese score is $i$ points, and mathematics score is $j$ points, let it correspond to the integer point $(i, j)$ on the plane, called a "score point". Thus, the examination results of $n$ students are mapped to $n$ score points within the range
$$
0 \leqslant x \leqslant 100,0 \leqslant y \leqslant 100
$$

on the plane.
Consider 201 lines on the plane:
$$
y=x \pm b(b=0,1, \cdots, 100) \text {. }
$$

If a line has three score points, it indicates that there are three students, A, B, and C, such that A is better than B, and B is better than C.

Clearly, the lines $y=x+100$ and $y=x-100$ can each have at most one score point; the lines $y=x+99$ and $y=x-99$ can each have at most two score points.

Since $2 \times(201-2)+1 \times 2=400$, when $n>400$, there must be a line with three score points.
Thus, the minimum value of $n$, $n_{0} \leqslant 401$.
Let the set
$S=\{(i, j) \mid i=0,1 ; j=0,1, \cdots, 100\}$;
$T=\{(i, j) \mid i=0,1, \cdots, 100 ; j=0,1\}$.
Clearly, $|S \cup T|=400$, and there do not exist three score points on the same line in $S \cup T$.
Therefore, $n_{0} \geqslant 401$.
Thus, $n_{0}=401$.