5$\cdot$102 In the parliament, there are 2000 deputies who decide to review the national budget, which consists of 200 expenditure items. Each deputy receives a draft of the budget and, based on their own judgment, lists the maximum limit for each item. However, the total sum of all amounts does not exceed the given total limit $S$. When the parliament reviews and determines the allocation for each item, it sets the amount to a figure that at least $k$ deputies agree on (i.e., at least $k$ deputies have given a maximum limit no less than this figure). What is the minimum value of $k$ that ensures the total approved allocation does not exceed the limit $S$?
The conclusion also holds.
If among the $n$ points there exists a point $A$ from which the number of line segments does not exceed $m$, then since there are a total of
$$
m n=m(k+1)=k m+m
$$

line segments, the remaining $k$ points must have at least $m k$ line segments among them. Thus, by the inductive hypothesis, the conclusion holds.

If among the $n$ points there is no point from which the number of line segments does not exceed $m$, i.e., each of the $n$ points has at least $m+1$ line segments, then we can construct the required sequence of points as follows: take any point $V_{0}$, then take any point $V_{1}$ that is connected to $V_{0}$. Since $m \geqslant 1$, there are still points connected to $V_{1}$ other than $V_{0}$, take one of these points as $V_{2}$, and continue this process until $V_{p}$.

If $p<m$, then $V_{p}$, apart from having at most $p$ connections to the previous $p$ points, still has connections to other points, so we can select $V_{p+1}$ that meets the requirements, leading to a contradiction.

Therefore, $p \geqslant m$, and thus we obtain the required sequence of points. Hence, when $n=k+1$, the conclusion holds. That is, for $n \geqslant 2 m+1$, the conclusion always holds.
[Solution] Divide 2000 senators into 200 groups, each with 10 people, and number these groups from 1 to 200. For each $i, 1 \leqslant i \leqslant 200$, the 10 people in the $i$-th group veto the $i$-th clause and agree to allocate $\frac{S}{199}$ for each of the remaining 199 clauses. If it is predetermined that $k=1990$, then the voting result will be that each clause receives an allocation of $\frac{S}{199}$, and the total allocation will exceed $S$. Therefore, $k=1990$ is insufficient, i.e., $k$ must be at least 1991.

On the other hand, when $k=1991$, for each expenditure, there are at most 9 senators whose proposed allocation limits are less than the approved amount. Therefore, for the 200 clauses, there is at least one clause for which the total number of senators whose proposed limits are less than the approved amount is less than 1800, meaning there is always one senator whose proposed limits for all clauses are not less than the approved amounts. Since the sum of the proposed limits does not exceed $S$, the total approved amount also does not exceed $S$, i.e., $k=1991$ is sufficient.
In summary, the minimum value of $k$ is 1991.