A new intersection theorem and its applications to bounding the chromatic numbers of spaces

In 1981 P. Frankl and R.M. Wilson showed that if H = ( V , E ) is a k-uniform hypergraph on n vertices and for any F 1 , F 2 ∈ E one has | F 1 ∩ F 2 | ≠ l with q = k − l being a prime power, then for 2 l < k , | E | ⩽ ∑ i = 0 q − 1 ( n i ) , and for 2 l ⩾ k , | E | ⩽ ( n d ) ( k d ) ∑ i = 0 q − 1 ( n − d i ) , where d = k − 2 q + 1 = 2 l − k + 1 . In this note, we improve the second inequality and find applications to coloring real and rational spaces.