THE EIGENVALUE METHOD FOR CROSS t-INTERSECTING FAMILIES

We show that the Erdős–Ko–Rado inequality for t-intersecting families of subsets can be easily extended to an inequality for cross t-intersecting families by using the eigenvalue method. The same applies to the case of t-intersecting families of subspaces. The eigenvalue method is one of the proof techniques to get Erdős–Ko–Rado type inequalities for t-intersecting families, for example, a proof for families of k-subsets by Wilson [9], a proof for families of k-subspaces by Frankl and Wilson [5], and a recent seminal proof for families of permutations by Ellis, Friedgut and Pilpel [3]. The last one contains a stronger inequality for cross t-intersecting families which follows from a variant of the Hoffman–Delsarte bound. In this note we remark that one can also get the corresponding cross t-intersecting version of the first two results about k-subsets (Theorem 1) and k-subspaces (Theorem 2) in the same way quite easily. Let Xn = {1,2, . . . ,n} be an n-element set. Two families of k-subsets A ,B ⊂ (Xn k ) are called cross t-intersecting if |A∩B| ≥ t holds for all A ∈ A ,B ∈ B. Theorem 1. Let n≥ k ≥ t ≥ 2 and −t log(1− k n) ≤ log2. Suppose that A ,B ⊂ (Xn k ) are cross t-intersecting. Then we have |A ||B| ≤ ( n− t k− t )2 . If |A ||B| = (n−t k−t )2, then A = B = {F ∈ ( Xn k ) : T ⊂ F} for some T ∈ (Xn t ) . Let Vn be an n-dimensional vector space over the q-element field. Let [Vn k ] denote the set of all k-subspaces (k-dimensional subspaces) of Vn, and let [n k ] = # [Vn k ] = ∏k−1 i=0 (q n−i− 1)/(qk−i−1). Two families of k-subspaces A ,B ⊂ [Vn k ] are called cross t-intersecting if dim(A∩B) ≥ t holds for all A ∈ A ,B ∈ B. Theorem 2. Let n≥ 2k≥ 2t. Suppose that A ,B ⊂ [Vn k ] are cross t-intersecting. Then we have |A ||B| ≤ [ n− t k− t ]2