134 research outputs found
The limit in the -Problem of Brown, Erd\H{o}s and S\'os exists for all
Let be the maximum number of edges of an -uniform
hypergraph on~ vertices not containing a subgraph with ~edges and at most
~vertices. In 1973, Brown, Erd\H{o}s and S\'os conjectured that the limit
exists for all positive integers
. They proved this for . In 2019, Glock proved this for and
determined the limit. Quite recently, Glock, Joos, Kim, K\"{u}hn, Lichev and
Pikhurko proved this for and determined the limit; we combine their work
with a new reduction to fully resolve the conjecture by proving that indeed the
limit exists for all positive integers .Comment: 10 pages, to appear in Proceedings of the AM
Finding an almost perfect matching in a hypergraph avoiding forbidden submatchings
In 1973, Erd\H{o}s conjectured the existence of high girth -Steiner
systems. Recently, Glock, K\"{u}hn, Lo, and Osthus and independently Bohman and
Warnke proved the approximate version of Erd\H{o}s' conjecture. Just this year,
Kwan, Sah, Sawhney, and Simkin proved Erd\H{o}s' conjecture. As for Steiner
systems with more general parameters, Glock, K\"{u}hn, Lo, and Osthus
conjectured the existence of high girth -Steiner systems. We prove the
approximate version of their conjecture.
This result follows from our general main results which concern finding
perfect or almost perfect matchings in a hypergraph avoiding a given set of
submatchings (which we view as a hypergraph where ). Our first
main result is a common generalization of the classical theorems of Pippenger
(for finding an almost perfect matching) and Ajtai, Koml\'os, Pintz, Spencer,
and Szemer\'edi (for finding an independent set in girth five hypergraphs).
More generally, we prove this for coloring and even list coloring, and also
generalize this further to when is a hypergraph with small codegrees (for
which high girth designs is a specific instance). Indeed, the coloring version
of our result even yields an almost partition of into approximate high
girth -Steiner systems.
Our main results also imply the existence of a perfect matching in a
bipartite hypergraph where the parts have slightly unbalanced degrees. This has
a number of applications; for example, it proves the existence of
pairwise disjoint list colorings in the setting of Kahn's theorem; it also
proves asymptotic versions of various rainbow matching results in the sparse
setting (where the number of times a color appears could be much smaller than
the number of colors) and even the existence of many pairwise disjoint rainbow
matchings in such circumstances.Comment: 52 page
On a Conjecture of Thomassen
In 1989, Thomassen asked whether there is an integer-valued function f(k)
such that every f(k)-connected graph admits a spanning, bipartite -connected
subgraph. In this paper we take a first, humble approach, showing the
conjecture is true up to a log n factor.Comment: 9 page
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A
family of permutations is \emph{-intersecting} if
any two permutations in agree on some indices, and is
\emph{trivial} if all permutations in agree on the same
indices. A -uniform hypergraph is \emph{-intersecting} if any two of its
edges have vertices in common, and \emph{trivial} if all its edges share
the same vertices.
The fundamental problem is to determine how large an intersecting family can
be. Ellis, Friedgut and Pilpel proved that for sufficiently large with
respect to , the largest -intersecting families in are the trivial
ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest
-intersecting -uniform hypergraphs are also trivial when is large. We
determine the \emph{typical} structure of -intersecting families, extending
these results to show that almost all intersecting families are trivial. We
also obtain sparse analogues of these extremal results, showing that they hold
in random settings.
Our proofs use the Bollob\'as set-pairs inequality to bound the number of
maximal intersecting families, which can then be combined with known stability
theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira
result. Update 2: corrected statement of the unpublished Hamm--Kahn result,
and slightly modified notation in Theorem 1.6 Update 3: new title, updated
citations, and some minor correction
Almost all 9-regular graphs have a modulo-5 orientation
In 1972 Tutte famously conjectured that every 4-edge-connected graph has a
nowhere zero 3-flow; this is known to be equivalent to every 5-regular,
4-edge-connected graph having an edge orientation in which every in-degree is
either 1 or 4. Jaeger conjectured a generalization of Tutte's conjecture,
namely, that every -regular, -edge-connected graph has an edge
orientation in which every in-degree is either or . Inspired by the
work of Pralat and Wormald investigating , for we show this holds
asymptotically almost surely for random 9-regular graphs. It follows that the
conjecture holds for almost all 9-regular, 8-edge-connected graphs. These
results make use of the technical small subgraph conditioning method.Comment: 18 page
Viewing extremal and structural problems through a probabilistic lens
This thesis focuses on using techniques from probability to solve problems from extremal and structural combinatorics. The main problem in Chapter 2 is determining the typical structure of -intersecting families in various settings and enumerating such systems. The analogous sparse random versions of our extremal results are also obtained. The proofs follow the same general framework, in each case using a version of the Bollobás Set-Pairs Inequality to bound the number of maximal intersecting families, which then can be combined with known stability theorems. Following this framework from joint work with Balogh, Das, Liu, and Sharifzadeh, similar results for permutations, uniform hypergraphs, and vector spaces are obtained.
In 2006, Barát and Thomassen conjectured that the edges of every planar 4-edge-connected 4-regular graph can be decomposed into disjoint copies of , the star with three leaves. Shortly afterward, Lai constructed a counterexample to this conjecture. Following joint work with Postle, in Chapter 3 using the Small Subgraph Conditioning Method of Robinson and Wormald, we find that a random 4-regular graph has an -decomposition asymptotically almost surely, provided we have the obvious necessary divisibility conditions.
In 1988, Thomassen showed that if is at least -edge-connected then has a spanning, bipartite -connected subgraph. In 1989, Thomassen asked whether a similar phenomenon holds for vertex-connectivity; more precisely: is there an integer-valued function such that every -connected graph admits a spanning, bipartite -connected subgraph? In Chapter 4, as in joint work with Ferber, we show that every -connected graph admits a spanning, bipartite -connected subgraph
Edge-colouring graphs with local list sizes
The famous List Colouring Conjecture from the 1970s states that for every
graph the chromatic index of is equal to its list chromatic index. In
1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds
asymptotically. Our main result is a local generalization of Kahn's theorem.
More precisely, we show that, for a graph with sufficiently large maximum
degree and minimum degree , the following
holds: for every assignment of lists of colours to the edges of , such that
for
each edge , there is an -edge-colouring of . Furthermore, Kahn
showed that the List Colouring Conjecture holds asymptotically for linear,
-uniform hypergraphs, and recently Molloy generalized Kahn's original result
to correspondence colouring as well as its hypergraph generalization. We prove
local versions of all of these generalizations by showing a weighted version
that simultaneously implies all of our results.Comment: 22 page
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