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In this note, we give short proofs of two theorems in extremal set theory. 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 upon 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.The second result partially proves a conjecture of Frankl and Tokushige about kuniform families with restricted pairwise intersection sizes.

Let µ 1 ≥ . . . ≥ µ n denote the eigenvalues of a graph G with m edges and clique number ω(G). Nikiforov proved a spectral version of Turán's theorem thatand Bollobás and Nikiforov conjectured that forThis paper proposes the conjecture that for all graphs (µ 2 1 + µ 2 2 ) in this inequality can be replaced by the sum of the squares of the ω largest eigenvalues, provided they are positive. We provide experimental and theoretical evidence for this conjecture, and describe how the bound can be applied.

The spread of a graph G is the difference between the largest and smallest eigenvalue of the adjacency matrix of G. Gotshall, O'Brien and Tait conjectured that for sufficiently large n, the nvertex outerplanar graph with maximum spread is the graph obtained by joining a vertex to a path on n − 1 vertices. In this paper, we disprove this conjecture by showing that the extremal graph is the graph obtained by joining a vertex to a path on ⌈(2n − 1)/3⌉ vertices and ⌊(n − 2)/3⌋ isolated vertices. For planar graphs, we show that the extremal n-vertex planar graph attaining the maximum spread is the graph obtained by joining two nonadjacent vertices to a path on ⌈(2n − 2)/3⌉ vertices and ⌊(n − 4)/3⌋ isolated vertices.

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