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up vote 5 down vote favorite 4 Consider the $35$ sets $A_1,A_2,dots,A_{35}$ such that $|A_i|=27$ for all $1leq i leq 35$ , and every triplet of sets have one exactly one element in common to all three. Prove that there is at least one element common to all $35$ sets. This was a problem given to me by a friend - he asked me to help him with this but I am unable to figure out the answer. He suggested something about contradiction and the pigeonhole principle, but I'm not sure how to continue. He mentioned it was from some sort of Olympiad, but I dont remember which one (something middle eastern?) I found this similar question online: https://artofproblemsolving.com/community/q1h1699161p10908761 but this doesn't include any restrictions on cardinality, and is only a case for a pairwise disjoint set.