Aimed at graduate students and researchers, this fascinating text provides a comprehensive study of the Erdős–Ko–Rado Theorem, with a focus on algebraic methods. The authors begin by discussing well-known proofs of the EKR bound for intersecting families. The natural generalization of the EKR Theorem holds for many different objects that have a notion of intersection, and the bulk of this book focuses on algebraic proofs that can be applied to these different objects. The authors introduce tools commonly used in algebraic graph theory and show how these can be used to prove versions of the EKR Theorem. Topics include association schemes, strongly regular graphs, the Johnson scheme, the Hamming scheme and the Grassmann scheme. Readers can expand their understanding at every step with the 170 end-of-chapter exercises. The final chapter discusses in detail 15 open problems, each of which would make an interesting research project.

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations π, σ in S there is a point i ∈ {1, . . . , n} such that π(i) = σ(i). Deza and Frankl [9] proved that if S ⊆ S(n) is intersecting then |S| ≤ (n − 1)!. Further, Cameron and Ku [4] show that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result. * Research supported by NSERC.The Erdős-Ko-Rado theorem [7] is a central result in extremal combinatorics. There are many interesting proofs and extensions of this theorem, for a summary see [6]. The Erdős-Ko-Rado theorem gives a bound on the size of a family of intersecting k-subsets of a set and describes exactly which families meet this bound.1.1 Theorem. (Erdős, Ko and Rado [7]) Let k, n be positive integers with n > 2k. Let A be a family of k-subsets of {1, . . . , n} such that any two sets from A have non-trivial intersection, then |A| ≤ (n − 1)!. Moreover, |A| = (n−1)! if and only if A is the collection of all k-subsets that contain a fixed i ∈ {1, . . . , n}.The Erdős-Ko-Rado theorem has been extended to objects other than subsets of a set. For example, Hsieh [17] and Frankl and Wilson [8] give a version for intersecting subspaces of a vector space over a finite field, Berge [3] proves it for intersecting integer sequences, Rands [21] extends it to intersecting blocks in a design and Meagher and Moura [19] prove a version for partitions.The extension we give here is to intersecting permutations. Let S(n) be the symmetric group on {1, . . . , n}. Permutations π, σ ∈ S(n) are said to be intersecting if π(i) = σ(i) for some i ∈ {1, . . . , n}. Similar to the case for subsets of a set, there are obvious candidates for maximum intersecting systems of permutations, these are the sets S i,j = {π ∈ S(n) : π(i) = j}, i, j ∈ {1, . . . , n}.(1.1)These sets are the cosets of a stabiliser of a point.1.2 Theorem. (Cameron and Ku [4]) Let n ≥ 2. If S ⊆ S(n) is an intersecting family of permutations then:(a) |S| ≤ (n − 1)!.(b) if |S| = (n − 1)! then S is a coset of a stabiliser of a point.The proof given by Cameron and Ku uses an operation called fixing which is similar to the shifting operation used in the original proof of Erdős-Ko-Rado. They show that a maximum intersecting family of permutations is closed under this fixing operation. Assuming that the family contains the identity permutation, and thus each permutation in the family has a fixed point, they next consider the set system formed by the sets of fixed points for each permutation in the family. Cameron and Ku prove that if the family of permutations is closed

In this paper we prove an Erdős-Ko-Rado-type theorem for intersecting sets of permutations. We show that an intersecting set of maximal size in the projective general linear group PGL 3 (q), in its natural action on the points of the projective line, is either a coset of the stabilizer of a point or a coset of the stabilizer of a line. This gives the first evidence to the veracity of [11, Conjecture 2].

Abstract. The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph G, is denoted by q(G). Using other parameters related to G, bounds for q(G) are proven and then applied to deduce further properties of q(G). It is shown that there is a great number of graphs G for which q(G) = 2. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs G are provided to show that adding and deleting edges or vertices can dramatically change the value of q(G). Finally, the set of graphs G with q(G) near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.

A subset S of a group G ≤ Sym(n) is intersecting if for any pair of permutations π, σ ∈ S there is an i ∈ {1, 2, . . . , n} such that π(i) = σ (i). It has been shown, using an algebraic approach, that the largest intersecting sets in each of Sym(n), Alt(n), and PGL(2, q) are exactly the cosets of the point-stabilizers. In this paper, we show how this approach can be applied more generally to many 2-transitive groups. We then apply this method to the Mathieu groups and to all 2-transitive groups with degree no more than 20.

Two set partitions of an $n$-set are said to $t$-intersect if they have $t$ classes in common. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of the Erdős-Ko-Rado theorem for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set. We prove that for $n$ large enough, any such system contains at most $${1\over(k-t)!} {n-tc \choose c} {n-(t+1)c \choose c} \cdots {n-(k-1)c \choose c}$$ partitions and this bound is only attained by a trivially $t$-intersecting system. We also prove that for $t=1$, the result is valid for all $n$. We conclude with some conjectures on this and other types of intersecting partition systems.

We prove an analogue of the classical Erdős-Ko-Rado theorem for intersecting sets of permutations in finite 2-transitive groups. Given a finite group G acting faithfully and 2-transitively on the set Ω, we show that an intersecting set of maximal size in G has cardinality |G|/|Ω|. This generalises and gives a unifying proof of some similar recent results in the literature.2010 Mathematics Subject Classification. Primary 05C35; Secondary 05C69, 20B05.

In this paper a new parameter for hypergraphs called hypergraph infection is defined. This concept generalizes zero forcing in graphs to hypergraphs. The exact value of the infection number of complete and complete bipartite hypergraphs is determined. A formula for the infection number for interval hypergraphs and several families of cyclic hypergraphs is given. The value of the infection number for a hypergraph whose edges form a symmetric t-design is given, and bounds are determined for a hypergraph whose edges are a t-design. Finally, the infection number for several hypergraph products and line graphs are considered.