25 research outputs found
The codegree threshold of
The codegree threshold of a -graph is the
minimum such that every -graph on vertices in which every pair
of vertices is contained in at least edges contains a copy of as a
subgraph. We study when , the -graph on
vertices with edges. Using flag algebra techniques, we prove that if is
sufficiently large then .
This settles in the affirmative a conjecture of Nagle from 1999. In addition,
we obtain a stability result: for every near-extremal configuration , there
is a quasirandom tournament on the same vertex set such that is close
in the edit distance to the -graph whose edges are the cyclically
oriented triangles from . For infinitely many values of , we are further
able to determine exactly and to show that
tournament-based constructions are extremal for those values of .Comment: 31 pages, 7 figures. Ancillary files to the submission contain the
information needed to verify the flag algebra computation in Lemma 2.8.
Expands on the 2017 conference paper of the same name by the same authors
(Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413
Covering and tiling hypergraphs with tight cycles
Given , we say that a -uniform hypergraph is a
tight cycle on vertices if there is a cyclic ordering of the vertices of
such that every consecutive vertices under this ordering form an
edge. We prove that if and , then every -uniform
hypergraph on vertices with minimum codegree at least has
the property that every vertex is covered by a copy of . Our result is
asymptotically best possible for infinitely many pairs of and , e.g.
when and are coprime.
A perfect -tiling is a spanning collection of vertex-disjoint copies
of . When is divisible by , the problem of determining the
minimum codegree that guarantees a perfect -tiling was solved by a
result of Mycroft. We prove that if and is not divisible
by and divides , then every -uniform hypergraph on vertices
with minimum codegree at least has a perfect
-tiling. Again our result is asymptotically best possible for infinitely
many pairs of and , e.g. when and are coprime with even.Comment: Revised version, accepted for publication in Combin. Probab. Compu
The Turán Density of Tight Cycles in Three-Uniform Hypergraphs
The Turán density of an -uniform hypergraph , denoted , is the limit of the maximum density of an -vertex -uniform hypergraph not containing a copy of , as . Denote by the -uniform tight cycle on vertices. Mubayi and Rödl gave an “iterated blow-up” construction showing that the Turán density of is at least , and this bound is conjectured to be tight. Their construction also does not contain for larger not divisible by , which suggests that it might be the extremal construction for these hypergraphs as well. Here, we determine the Turán density of for all large not divisible by , showing that indeed . To our knowledge, this is the first example of a Turán density being determined where the extremal construction is an iterated blow-up construction. A key component in our proof, which may be of independent interest, is a -uniform analogue of the statement “a graph is bipartite if and only if it does not contain an odd cycle”