Note on set systems without a strong simplex

A d -simplex is a collection of d + 1 sets such that every d of them have nonempty intersection and the intersection of all of them is empty. A strong d -simplex is a collection of d + 2 sets A , A 1 , … , A d + 1 such that { A 1 , … , A d + 1 } is a d -simplex, while A contains an element of ∩ j ≠ i A j for each i , 1 ≤ i ≤ d + 1 . ≥ d + 1 ≥ 3 and n > k ( d + 1 ) / d . It was conjectured by Mubayi and Ramadurai that if F is a collection of k -element subsets of an n -element set that contains no strong d -simplex, then | F | ≤ n − 1 k − 1 with equality only when F is a star. We prove their conjecture when k = d + 1 . This also gives a strengthening of a result of Chvátal on set systems without a simplex.