KöMaL 2026 C1903

The 17 enthusiastic members of a math club would like to get together for a group math session during the summer break, so they collect everyone's availability: from each person, they receive a time interval consisting of at least two days. (For example, from July 3 to July 6.) After collecting the data, it turned out that among any three students there are two who have marked a common day. Is it guaranteed that there will be a day such that every student will be available? Is it guaranteed that there will be two days such that every student will be available on at least one of the two days?