This problem could be simply solved for Greedy. Each time, we choose a new event with the smallest start day. If there are multiple event with this start day, we choose the one with the smallest end day.

```
import heapq
class Solution:
def maxEvents(self, events: List[List[int]]) -> int:
heap = []
for event in events:
heapq.heappush(heap, event)
current = 0
count = 0
while heap:
event = heapq.heappop(heap)
if current >= event[1]:
continue
elif event[0] < current + 1:
event[0] = current + 1
heapq.heappush(heap, event)
else:
current = event[0]
count += 1
return count
```