Finding Conflict algorithm doubt Let $e_1,e_2,\cdots,e_n$ are some events are given by their starting time and ending time. 

I have to find an event that conflict with maximum number of other events. 

Conflicts means interval overlaps

The idea is as follows: Sort events according to their starting times and then for each event check whether it conflict with other evenets or not. The first step seems to me doable in $\mathcal{O}(n \log n)$ time, but second step i.e. to find the number of events with which it conflicts is seems to $\mathcal{O}(n)$ Can we do it in $\mathcal{O}(\log n)$ time?

Question : Is it possible to do the second step as mentioned above in $\mathcal{O}(\log n)$ time?
Take all $2n$ starting and ending times and sort them, then walk through the list in sorted order, increasing a counter at each "starting" time, and decreasing the counter at each "ending" time.  On every counter increase, check whether this is the highest the counter has gone so far, and if so update the best-so-far event.