49 research outputs found
Balanced supersaturation for some degenerate hypergraphs
A classical theorem of Simonovits from the 1980s asserts that every graph
satisfying must contain copies of . Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
has copies of , which
are `uniformly distributed' over the edges of . Moreover, they used this
result to obtain a sharp bound on the number of -free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to -graphs. We also prove analogous results for complete
-partite -graphs.Comment: Changed title, abstract and introduction were rewritte
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 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