Short proofs of three results about intersecting systems

In this note, we give short proofs of three theorems about intersection problems.The first one is a determination of the maximum size of a nontrivial k-uniform, d-wise intersecting family for n ⩾ 1 + d 2 (k-d+2), which improves the range of n of a recent result of O'Neill and Verstraëte.Our proof also extends to d-wise, t-intersecting families, and from this result we obtain a version of the Erdős-Ko-Rado theorem for d-wise, t-intersecting families.Our second result partially proves a conjecture of Frankl and Tokushige about k-uniform families with restricted pairwise intersection sizes.Our third result is about intersecting families of graphs.Answering a question of Ellis, we construct K s,t -intersecting families of graphs which have size larger than the Erdős-Ko-Rado-type construction, whenever t is sufficiently large in terms of s.The construction is based on nontrivial (2s)-wise t-intersecting families of sets.