657 research outputs found
Short proofs of some extremal results
We prove several results from different areas of extremal combinatorics,
giving complete or partial solutions to a number of open problems. These
results, coming from areas such as extremal graph theory, Ramsey theory and
additive combinatorics, have been collected together because in each case the
relevant proofs are quite short.Comment: 19 page
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
Toric rings, inseparability and rigidity
This article provides the basic algebraic background on infinitesimal
deformations and presents the proof of the well-known fact that the non-trivial
infinitesimal deformations of a -algebra are parameterized by the
elements of cotangent module of . In this article we focus on
deformations of toric rings, and give an explicit description of in
the case that is a toric ring.
In particular, we are interested in unobstructed deformations which preserve
the toric structure. Such deformations we call separations. Toric rings which
do not admit any separation are called inseparable. We apply the theory to the
edge ring of a finite graph. The coordinate ring of a convex polyomino may be
viewed as the edge ring of a special class of bipartite graphs. It is shown
that the coordinate ring of any convex polyomino is inseparable. We introduce
the concept of semi-rigidity, and give a combinatorial description of the
graphs whose edge ring is semi-rigid. The results are applied to show that for
, is not rigid while for , is
rigid. Here is the complete bipartite graph with one
edge removed.Comment: 33 pages, chapter 2 of the Book << Multigraded Algebra and
Applications>> 2018, Springer International Publishing AG, part of Springer
Natur
Saturation numbers in tripartite graphs
Given graphs and , a subgraph is an -saturated
subgraph of if , but for all . The saturation number of in , denoted
, is the minimum number of edges in an -saturated subgraph
of . In this paper we study saturation numbers of tripartite graphs in
tripartite graphs. For and , , and sufficiently
large, we determine and
exactly and
within an additive constant.
We also include general constructions of -saturated subgraphs of
with few edges for .Comment: 18 pages, 6 figure
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