A Vector Space Analog of Lovasz s Version of the Kruskal-Katona Theorem

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1 Claude Tardif Non-canonical Independent sets in Graph Powers Let s 4 be an integer. The truncated s-simplex T s is defined as follows: V (T s ) = {(i, j) {0, 1,..., s 1} 2 : i j}, E(T s ) = {[(i, j), (, l)] : i =, j l or i = l, j = }. The truncated simplices are vertex-transitive, and it is nown that their categorical powers have maximum independent sets that are non-canonical, in the sense that they are not inverse images of an independent set under a projection. Question: Are there other families of vertex-transitive graphs whose powers have non-canonical maximum independent sets? In particular, the core of T s is the clique K s 1 (by Broo s theorem), so families of core examples would be interesting. In contrast, recent research on Fourier analysis and spectral techniques has outlined sufficient conditions for a graph to have only canonical maximum independent sets in its powers. Ameera Chowdhury anchowdh@math.ucsd.edu A Vector Space Analog of Lovasz s Version of the Krusal-Katona Theorem Let V denote a n-dimensional vector space over a finite field of order q. For Z, we write [ ] V to denote the family of all -dimensional subspaces q of V. For a R and Z, define the Gaussian binomial coefficient by [ ] a := q a i 1 q i 1. If and q are fixed, then [ ] a q q 0 i< is a continuous function of a and is positive and strictly increasing when a. From now on, we will omit the subscript q. We define the shadow of F [ V, denoted F, to consist of those ( 1)- dimensional subspaces of V contained in at least one member of F; that 1

2 is, F := { E [ V 1] : E F F }. Balász Patós and I recently proved the following vector space analog of Lovasz s version of the Krusal-Katona theorem. Theorem 1. Suppose F [ V. Let y be the real number defined by F = [ ] [ y. Then F y ] 1. If equality holds, then y Z and F = [ Y, where Y is a y-dimensional subspace of V. In extremal set theory, Lovasz s version of the Krusal-Katona theorem has many applications. For example, Dayin essentially used Lovasz s theorem to give a proof of the Erdős-Ko-Rado theorem. We will call a family ] r-wise intersecting if for all F1,..., F r F we have r i=1 F i {0}. F [ V Balász Patos and I applied Theorem 1 to give an upper bound on r-wise intersecting families in vector spaces. Theorem 2. Suppose F [ V is r-wise intersecting and r (r 1)n. Then F [ { [ n 1 1]. Moreover, equality holds if and only if F = F V ] } : v F for some one-dimensional subspace v V, unless r = 2 and n = 2. Observe that the case r = 2 of Theorem 2 is the Erdős-Ko-Rado theorem for vector spaces. In the language of graph theory, the Erdős-Ko-Rado theorem for vector spaces gives the size and structure of a maximum coclique in the so-called q-kneser graph; the q-kneser graph has the -dimensional subspaces of V as its vertices, where two subspaces α, β are adjacent if α β = {0}. It would be nice to discover new applications of Theorem 1. For example, does Theorem 1 imply anything about t-intersecting or cross-intersecting families in vector spaces? Cheng Yeaw Ku Bounds on the Eigenvalues of the Derangement Graph Let Γ n denote the Cayley graph on the symmetric group S n generated by the set D n of derangements (fixed-point free elements). It is well nown that the eigenvalues of Γ n are integers given by η χ = 1 χ(1) 2 s D n χ(s),

3 where χ ranges over all the irreducible characters of S n. Moreover, the irreducible characters of S n are indexed by partitions λ of n. We write η λ to denote the eigenvalue η χλ of Γ n, where χ λ is the irreducible character indexed by the partition λ n. Conjecture 1. [1] Suppose λ n is the largest partition in lexicographic order among all the partitions with λ 1 as their first part. Then, for every λ = (λ 1,..., λ s ) n, η (λ1,1 n λ 1 ) η λ η λ. References [1] C. Y. Ku and D. B. Wales, The eigenvalues of the derangement graph, Journal of Combinatorial Theory Series A, article in press. Reza Naserasr A Conjecture of Alon-Sas-Seymour: Conjecture 2. For a graph G, if the edges of G can be partitioned into n complete bipartite graphs then G is n 1 colourable. We have concrete examples that might be counterexamples to this. Bill Martin Delsarte s LP bound v.s. Hoffman bound Consider a symmetric association scheme with associate matrices {A 0, A 1, A 2,...}. When is the optimum solution to Delsarte s LP for a {1, 2}-code better than the Hoffman bound? Uniform one-factorizations of the complete bipartite graph Let {R [1 n ], R λ1, R λ2,..., R [n] } (where [1 n ], λ 1, λ 2,..., [n] are all the integer partitions of n) be the group association scheme (also called the conjugacy class scheme) of the symmetric group S n. For which integer partitions λ of n, does there exist a clique of size n in the graph R λ? 3

4 This was a social research project of mine bac in the mid-90s, with anyone interested welcome to mae contributions. Gillian Nonay and I were the main players, but there were contributions from Dan Archdeacon, Gordon Royle and others. I have a web page on the subject ( martin/research/onefac.html). The case σ = [n] is well-studied in graph theory: these are called perfect one-factorizations the union of any two of the matchings is a Hamilton cycle. Linear programming seems to be vacuous in this situation, so other non-existence results are needed. Karen Meagher meagher@math.uregina.ca EKR Theorem for Partitions Let P (, ) denote the collection of all set partitions of {1,..., 2 } into parts, each part of size. Two partitions P and Q from P (, ) are intersecting if there is a part of P and a part of Q such that the size of the intersection of these two part is at least 2. What is the largest collection of partitions from P (, ) such that any two partitions from the collection are intersecting.? There are obvious candidates for the largest such collection. Let i, j be distinct values in {1,..., 2 }. Let S i,j be the collection of all partitions from P (, ) that have a part that contains both i and j. This problem can be rephased as a question about the maximum independent sets in a graph. Consider a graph whose vertex set is the set of partitions from P (, ) and vertices are adjacent if and only if they are sew, that is, any two cells have intersection of size exactly 1. Call this the partition graph. An independent set in the partition graph is a collection of interseting partitions. If is a prime power then there is a clique of size 1 in the partition graph (such a clique is equivalent to an orthogonal array). In this case, from the clique/coclique bound we now that the sets S i,j are maximum cliques. It would be nice to show that this is true for values of and further to show that the sets S i,j are all the maximum independent sets in the partition graph. Another interesting problem is to find the least eigenvalue for the partition graph. It is not difficult to show that! 1 is an eigenvalue of the partition graph. In the case where is a prime power, from the ratio bound for cliques, we now that! 1 is the least eigenvalue. I conjecture that this is 4

5 the least eigenvalue for all values of. If it can be shown that this is the least eignevalue, then using the ratio bound for independent set, the collections S i,j must be the largest independent sets in the partition graph. Two final questions are: what is the chromatic number of the partition graph and is the partition graph a core? Induced Bipartite Subgraphs in the Kneser Graph What is the largest induced bipartite subgraph in the Kneser graph graph K(2 1, ) (or more generally K(n, ))? It is possible to construct an induced bipartite subgraph by taing one bipartition to be all the subsets that contain 1 and the other bipartition to be all the subsets that contain 2, but not 1. But, in the graph K(2 1, ), it is not hard to construct a larger induced bipartite subgraph. For example, if = 4 the size of the bipartite subgraph induced by the collection of sets that contain 1 and the sets that contain the element 2 but not 1 is ( ) ( ) 8 7 = Next let A be the collection of sets in V (K(9, 4)) that have at least 2 of the 3 elements {1, 2, 3} and let B be the collection of sets in V (K(9, 4)) that have at least 3 of the 5 elements {5, 6, 7, 8, 9}. The set A and B induce a bipartite subgraph of K(9, 4) with size (( )( ) ( )( )) (( )( ) ( )( )) = 96. This induced subgraph is conjectured to be the largest. With the interia bound it can be shown that the size of the largest induced subgraph in K(9, 4) is no larger than 98. 5