Why is it always optimal to choose the latest person available? I cannot understand why that is optimal (Why can't we remove the element at the 0th place in case of a tie)?

multiset<int> ms
it = ms.upper_bound(v[i]) 
if (it != ms.begin()) continue

Formally, why ms.erase(--it) instead of ms.erase(ms.begin()) ??

It can be easier like this:

    int n, k;
    cin >> n >> k;
    vector<pair<int, int>> a(n);
    for (auto &[x, y] : a) cin >> y >> x;
    sort(a.begin(), a.end());
    multiset<int> S;
    for (int i = 0; i < k; i ++ ) S.insert(0);
    int res = 0;
    for (auto &[y, x] : a)
    {
        auto it = S.upper_bound(x);
        if (it == S.begin()) continue;
        -- it;
        S.erase(it);
        S.insert(y);
        res ++ ;
    }
    cout << res << '\n';