A football league has $18$ teams competing against each other, each team has $8$ matches with $8$ other teams. At this time, Have or have not $3$ teams have no match with each other?
This problem was invented based on a Russian math. Then I do not know where to start to solve this problem.
You don't know. To see that you can have three teams that do not play each other, divide the teams into six groups of three, then put three groups into each each of two pools. The teams in each group do not play each other, but play all the teams in the other two groups in the pool. That gets each team six games. Mow match each group with a group from the other pool. Each team plays two of the three in the other group. This gives you six teams such that no two play each other, three from one of the first groups and three from one of the second groups.
To see you can have a tournament such that there are no three teams that do not play each other, divide the teams into two groups of nine. Let each group of nine play a round robin. Any set of three teams has two that are in the same group of nine and have played each other.