The International games comes usually in batches of 4, with 8 teams fighting each other in "bests of one".
So even if you are watching 1/4 of the games available, you are still going to watch more than 7 games in a row if you want to see all which is happening in that day, and there is more tomorrow.
Of course, like the olympics, nobody expect you to see it ALL but damn if people won't try.
Personally I'm just doing a mix of following the teams I really like and checking the games people say are a must watch after the match is done.
Tomorrow matches will be even better since lots of teams will be on the chopping block which will make for some tense games and some big names may end up going home early.
It's a good day to be a Dota fan.