Belarus2022

Answer: $2020$.

Let Dima wrote down the numbers $d_1>d_2>\cdots >d_{2022}$ and Vlad wrote down the numbers $v_1>v_2>\cdots >v_{2022}$. Suppose the answer in the problem is $2022$ or $2021$, then $d_{2022}>v_2$. Each of the $2021$ lines not containing $v_1$ contains at least $2021$ numbers not exceeding $v_2$: the number written by Vlad and all smaller numbers. And each column contains at least $2$ numbers not less than $d_{2022}$: the number written by Dima and all greater numbers. Therefore the total amount of numbers in the table is not less than $$2021\cdot 2021+2022\cdot 2=2022\cdot 2022+1,$$ which exceeds the number of all numbers in the table — a contradiction.

Let's show that it could turn out that $2020$ of numbers written by Vlad are less than any number written by Dima. Fill in the table

$v_1$	$d_1$	$\infty$	$d_4$	$d_5$	$d_6$	...	$d_{2022}$
$d_2$	$v_2$	$d_3$	$\infty$	$y$	$y$	...	$y$
$x$		$v_3$
$x$			$v_4$
$x$				$v_5$
$x$					$v_6$
$⋮$						$⋮$
$x$					$v_6$		$v_{2022}$

numbers from $1$ to $2022^2$ so that $$\infty >v_1>v_2>y>d_1>\cdots >d_{2022}>x>v_3>\cdots >v_{2020}$$ (by $\infty$, $y$ and $x$ in these inequalities we mean each number written on the cell with the corresponding label), and the numbers on empty cells are less than each of the numbers on marked cells.