50 research outputs found
Multipartite hypergraphs achieving equality in Ryser's conjecture
A famous conjecture of Ryser is that in an -partite hypergraph the
covering number is at most times the matching number. If true, this is
known to be sharp for for which there exists a projective plane of order
. We show that the conjecture, if true, is also sharp for the smallest
previously open value, namely . For , we find the minimal
number of edges in an intersecting -partite hypergraph that has
covering number at least . We find that is achieved only by linear
hypergraphs for , but that this is not the case for . We
also improve the general lower bound on , showing that .
We show that a stronger form of Ryser's conjecture that was used to prove the
case fails for all . We also prove a fractional version of the
following stronger form of Ryser's conjecture: in an -partite hypergraph
there exists a set of size at most , contained either in one side of
the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac
Forcing large tight components in 3-graphs
Any -vertex -graph with minimum codegree at least
must have a spanning tight component, but immediately below this threshold it
is possible for no tight component to span more than
vertices. Motivated by this observation, we ask which codegree forces a tight
component of at least any given size. The corresponding function seems to have
infinitely many discontinuities, but we provide upper and lower bounds, which
asymptotically converge as the function nears the origin.Comment: 10 pages. Final version accepted by European J. Combi
HIGHER-DIMENSIONAL HIGHLY CONNECTED RAMSEY THEORY (Set Theory : Reals and Topology)
Bergfalk, Hrusak, and Shelah recently introduced a weakening of the classical partition relation for pairs in which the complete monochromatic subgraph of the classical relation is replaced by a highly connected monochromatic subgraph. In subsequent work, we proved that, assuming the consistency of the existence of a weakly compact cardinal, it is consistent that an optimal square-bracket version of this highly connected partition relation holds at the continuum. In this paper, we introduce a higher-dimensional generalization of the highly connected partition relation and prove an analogous consistency result indicating that, if the existence of a weakly compact cardinal is consistent, then it is consistent that an optimal square-bracket version of the higherdimensional highly connected partition relation holds at the continuum
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics