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COMBINATORICS

DISJOINT REPRESENTABILITY OF SETS

JÓZSEF BALOGH

Abstract

For a hypergraph $\mathcal{H}$ and a set $S$, the trace of $\mathcal{H}$ on $S$ is the set of all intersections of edges of $\mathcal{H}$ with $S$. We will consider forbidden trace problems, in which we want to find the largest hypergraph $\mathcal{H}$ that does not contain some list of forbidden configurations as traces, possibly with some restriction on the number of vertices or the size of the edges in $\mathcal{H}$. Write $[k]^{(\ell)}$ for the set of all $\ell$-subsets of $[k]=\{1,\cdots,k\}$. Note that $\mathcal{A}$ has $k$ disjointly representable sets exactly when it has a $[k]^{(1)}$ trace. We will focus on three forbidden configurations: the $k$-singleton $[k]^{(1)}$, the $k$-co-singleton $[k]^{(k-1)}$ and the $k$-chain $\mathcal{C}_k = \{ \emptyset, \{1\}, [1,2], \cdots , [1,k-1] \}$. We prove a number of results on the size of the largest hypergraph $\mathcal{H}$ with some combination of these traces forbidden, sometimes with restrictions on the number of vertices or the size of the edges. We obtain exact results in the case $k=3$, both for uniform and non-uniform hypergraphs, and classify the extremal examples, and asymptotical results for larger values of $k$.

This is joint work with P. Keevash and B. Sudakov.

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POLYGONAL GRAPHS

CAI-HENG LI

Abstract

A near-polygonal graph is a graph $\Gamma$ which has a set $\mathcal C$ of $m$-cycles for some positive integer $m$ such that each $2$-path of $\Gamma$ is contained in exactly one cycle in $\mathcal C$. If $m$ is the girth of $\Gamma$ then the graph is called polygonal. We introduce a method for constructing near-polygonal graphs with $2$-arc transitive automorphism groups. As special cases, we obtain several new infinite families of polygonal graphs.

This is joint work with Ákos Seress.

THE INVARIANT FACTORS OF THE INCIDENCE

MATRICES OF POINTS AND LINEAR SUBSPACES

IN $PG(n,q)$ AND $AG(n,q)$

QING XIANG

Abstract

(Joint work with David B. Chandler and Peter Sin)

Let $V$ be an $(n+1)$-dimensional vector space over ${\rm GF}(q)$, where $q=p^t$, $p$ is a prime. For $1<r\leq n$, let $A_{1,r}^n(q)$ be the (0,1)-incidence matrix with rows and columns respectively indexed by the $r$- and $1$-dimensional subspaces of $V$, and with $(X,Y)$-entry equal to one if and only if the 1-dimensional subspace $Y$ is contained in the $r$-dimensional subspace $X$. The $p$-rank of $A_{1,r}^n(q)$ was computed by Smith and Hamada in late 1960s. In this talk, we explain how to determine the Smith normal form of $A_{1,r}^n(q)$. The techniques we used are from number theory ($p$-adic estimates of multiplicative character sums, Jacobi sums) and representations of the general linear groups.

GROUP THEORY

(to be announced)

MICHAEL COLLINS, Oxford & OSU

Abstract

We locate characteristic $p$-subgroups of quasiprimitive linear groups which behave very much like (quasisimple) components in arbitrary finite groups and which are central to establishing precise bounds for Jordan's theorem (apres Weisfeiler).

PARALLELISM, THE COSET LATTICE AND

AUTOPROJECTIVITIES OF FINITE ABELIAN $p$-GROUPS.

AN EXPOSITORY PRESENTATION

CHARLES HOLMES, Miami, Ohio

Abstract

This talk is three separate, but related, observations. Let $F$ be a field and $V$ be a vector space of dimension $n$ over $F$. The set of all cosets $u+S$ of all subspaces $S$ in $V$ is said to be an affine geometry. The dimension of a coset $u+S$ is the dimension of its associated vector subspace $S$. Two cosets $u+S$ and $v+T$ can be said to be parallel if and only if $S$ is a linear subspace of $T$ or $T$ is a subspace of $S$, [Snapper and Troyer]. The fact that two cosets of any subgroup of a group either are the same or disjoint gives the following characterization of parallel. Result 1. Two affine subspaces $u+S$ and $v + S$ are either identically equal or disjoint.

The second observation is that the set of all cosets, right or left, of a group together with the empty set forms a lattice, R. Schmidt. The meet of any two cosets is their intersection and the join of any two cosets is the meet of all cosets containing the given cosets. The lattice of all subgroups of G is an interval in this lattice of cosets. The lattice of cosets generalizes affine geometry in about the same way that the lattice of subgroups generalizes projective geometry. In particular, parallel cosets are simply disjoint.

The third observation is about the paper on Autoprojectivities of Abelian $p$-groups that Zacher, Costantini, and I published in 1998. The heavy lifting in this 1998 paper depends on proofs of results about an associated lattice of cosets. The main point of this talk is to make the coset lattice arguments more familiar and to wonder if we can do the proof with subgroups alone.