279 research outputs found
Tournaments, 4-uniform hypergraphs, and an exact extremal result
We consider -uniform hypergraphs with the maximum number of hyperedges
subject to the condition that every set of vertices spans either or
exactly hyperedges and give a construction, using quadratic residues, for
an infinite family of such hypergraphs with the maximum number of hyperedges.
Baber has previously given an asymptotically best-possible result using random
tournaments. We give a connection between Baber's result and our construction
via Paley tournaments and investigate a `switching' operation on tournaments
that preserves hypergraphs arising from this construction.Comment: 23 pages, 6 figure
EXTENDING POTOÄŒNIK AND Å AJNA'S CONDITIONS ON VERTEX-TRANSITIVE SELF-COMPEMENTARY k-HYPERGRAPHS
Abstract : Let l be a positive integer, k = 2l or k = 2l + 1 and let n be a positive integer with (mod ). Potocnik and Sajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have (where denotes the largest integer for which divides ). Here we extend their result to any integer k and a larger class of integers n
On the number of 4-cycles in a tournament
If is an -vertex tournament with a given number of -cycles, what
can be said about the number of its -cycles? The most interesting range of
this problem is where is assumed to have cyclic triples for
some and we seek to minimize the number of -cycles. We conjecture that
the (asymptotic) minimizing is a random blow-up of a constant-sized
transitive tournament. Using the method of flag algebras, we derive a lower
bound that almost matches the conjectured value. We are able to answer the
easier problem of maximizing the number of -cycles. These questions can be
equivalently stated in terms of transitive subtournaments. Namely, given the
number of transitive triples in , how many transitive quadruples can it
have? As far as we know, this is the first study of inducibility in
tournaments.Comment: 11 pages, 5 figure
Self-Complementary Hypergraphs
In this thesis, we survey the current research into self-complementary hypergraphs,
and present several new results.
We characterize the cycle type of the permutations on n elements with order equal
to a power of 2 which are k-complementing. The k-complementing permutations map
the edges of a k-uniform hypergraph to the edges of its complement. This yields a test
to determine whether a finite permutation is a k-complementing permutation, and
an algorithm for generating all self-complementary k-uniform hypergraphs of order
n, up to isomorphism, for feasible n. We also obtain an alternative description of
the known necessary and sufficient conditions on the order of a self-complementary
k-uniform hypergraph in terms of the binary representation of k.
We examine the orders of t-subset-regular self-complementary uniform hyper-
graphs. These form examples of large sets of two isomorphic t-designs. We restate
the known necessary conditions on the order of these structures in terms of the binary
representation of the rank k, and we construct 1-subset-regular self-complementary
uniform hypergraphs to prove that these necessary conditions are sufficient for all
ranks k in the case where t = 1.
We construct vertex transitive self-complementary k-hypergraphs of order n for
all integers n which satisfy the known necessary conditions due to Potocnik and Sajna, and consequently prove that these necessary conditions are also sufficient. We
also generalize Potocnik and Sajna's necessary conditions on the order of a vertex transitive self-complementary uniform hypergraph for certain ranks k to give neces-
sary conditions on the order of these structures when they are t-fold-transitive. In
addition, we use Burnside's characterization of transitive groups of prime degree to
determine the group of automorphisms and antimorphisms of certain vertex transitive self-complementary k-uniform hypergraphs of prime order, and we present an
algorithm to generate all such hypergraphs.
Finally, we examine the orders of self-complementary non-uniform hypergraphs,
including the cases where these structures are t-subset-regular or t-fold-transitive. We find necessary conditions on the order of these structures, and we present constructions to show that in certain cases these necessary conditions are sufficient.University of OttawaDoctor of Philosophy in Mathematic
Induced subgraph density. VI. Bounded VC-dimension
We confirm a conjecture of Fox, Pach, and Suk, that for every , there
exists such that every -vertex graph of VC-dimension at most has a
clique or stable set of size at least . This implies that, in the language
of model theory, every graph definable in NIP structures has a clique or
anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko,
and Thomas.
Our result also implies that every two-colourable tournament satisfies the
tournament version of the Erd\H{o}s-Hajnal conjecture, which completes the
verification of the conjecture for six-vertex tournaments. The result extends
to uniform hypergraphs of bounded VC-dimension as well.
The proof method uses the ultra-strong regularity lemma for graphs of bounded
VC-dimension proved by Lov\'asz and Szegedy and the method of iterative
sparsification introduced by the authors in an earlier paper.Comment: 11 pages, minor revision
Hypergraph Learning with Line Expansion
Previous hypergraph expansions are solely carried out on either vertex level
or hyperedge level, thereby missing the symmetric nature of data co-occurrence,
and resulting in information loss. To address the problem, this paper treats
vertices and hyperedges equally and proposes a new hypergraph formulation named
the \emph{line expansion (LE)} for hypergraphs learning. The new expansion
bijectively induces a homogeneous structure from the hypergraph by treating
vertex-hyperedge pairs as "line nodes". By reducing the hypergraph to a simple
graph, the proposed \emph{line expansion} makes existing graph learning
algorithms compatible with the higher-order structure and has been proven as a
unifying framework for various hypergraph expansions. We evaluate the proposed
line expansion on five hypergraph datasets, the results show that our method
beats SOTA baselines by a significant margin
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
A simultaneous generalization of independence and disjointness in boolean algebras
We give a definition of some classes of boolean algebras generalizing free
boolean algebras; they satisfy a universal property that certain functions
extend to homomorphisms. We give a combinatorial property of generating sets of
these algebras, which we call n-independent. The properties of these classes
(n-free and omega-free boolean algebras) are investigated. These include
connections to hypergraph theory and cardinal invariants on these algebras.
Related cardinal functions, Ind, which is the supremum of the cardinalities
of n-independent subsets; i_n, the minimum size of a maximal n-independent
subset; and i_omega, the minimum size of an omega-independent subset, are
introduced and investigated. The values of i_n and i_omega on P(omega)/fin are
shown to be independent of ZFC.Comment: Sumbitted to Orde
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