Geodesic
Intersection Theorems for Systems of Sets
Canadian mathematical bulletin, Tome 20 (1977) no. 2, pp. 249-254

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Let n and k be positive integers, k≥3. Denote by φ(n, k) the least positive integer such that if F is any family of more than φ(n, k) sets, each set with n elements, then some k members of F have pairwise the same intersection. In this paper we obtain a new asymptotic upper bound for φ(n, k), k fixed, n approaching infinity. 
Spencer, Joel. Intersection Theorems for Systems of Sets. Canadian mathematical bulletin, Tome 20 (1977) no. 2, pp. 249-254. doi: 10.4153/CMB-1977-038-7
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     author = {Spencer, Joel},
     title = {Intersection {Theorems} for {Systems} of {Sets}},
     journal = {Canadian mathematical bulletin},
     pages = {249--254},
     year = {1977},
     volume = {20},
     number = {2},
     doi = {10.4153/CMB-1977-038-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-038-7/}
}
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