1,620 research outputs found
Lorentz and Gale-Ryser theorems on general measure spaces
Based on the Gale-Ryser theorem for the existence of suitable
-matrices for different partitions of a natural number, we revisit the
classical result of G. G. Lorentz regarding the characterization of a plane
measurable set, in terms of its cross sections, and extend it to general
measure spaces
Lorentz and Gale-Ryser theorems on general measure spaces
Based on the Gale–Ryser theorem [2, 6], for the existence of suitable (0,1) -matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.Peer ReviewedPostprint (author's final draft
A family of extremal hypergraphs for Ryser's conjecture
Ryser's Conjecture states that for any -partite -uniform hypergraph,
the vertex cover number is at most times the matching number. This
conjecture is only known to be true for in general and for
if the hypergraph is intersecting. There has also been considerable effort made
for finding hypergraphs that are extremal for Ryser's Conjecture, i.e.
-partite hypergraphs whose cover number is times its matching number.
Aside from a few sporadic examples, the set of uniformities for which
Ryser's Conjecture is known to be tight is limited to those integers for which
a projective plane of order exists.
We produce a new infinite family of -uniform hypergraphs extremal to
Ryser's Conjecture, which exists whenever a projective plane of order
exists. Our construction is flexible enough to produce a large number of
non-isomorphic extremal hypergraphs. In particular, we define what we call the
{\em Ryser poset} of extremal intersecting -partite -uniform hypergraphs
and show that the number of maximal and minimal elements is exponential in
.
This provides further evidence for the difficulty of Ryser's Conjecture
Non-intersecting Ryser hypergraphs
A famous conjecture of Ryser states that every -partite hypergraph has
vertex cover number at most times the matching number. In recent years,
hypergraphs meeting this conjectured bound, known as -Ryser hypergraphs,
have been studied extensively. It was recently proved by Haxell, Narins and
Szab\'{o} that all -Ryser hypergraphs with matching number are
essentially obtained by taking disjoint copies of intersecting -Ryser
hypergraphs. Abu-Khazneh showed that such a characterisation is false for by giving a computer generated example of a -Ryser hypergraph with whose vertex set cannot be partitioned into two sets such that we have an
intersecting -Ryser hypergraph on each of these parts. Here we construct new
infinite families of -Ryser hypergraphs, for any given matching number , that do not contain two vertex disjoint intersecting -Ryser
subhypergraphs.Comment: 8 pages, some corrections in the proof of Lemma 3.6, added more
explanation in the appendix, and other minor change
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