499 research outputs found
Constructing Regular Self-complementary Uniform Hypergraphs
AMS Subject Classication Codes: 05C65, 05B05 05E20, 05C85.In this paper, we examine the possible orders of t-subset-regular self-complementary k-uniform hypergraphs, which form examples of large sets of two isomorphic t-designs. We reformulate Khosrovshahi and Tayfeh-Rezaie's necessary conditions on the order of these structures in terms of the binary representation of the rank k, and these conditions simplify to a more transparent relation between the order n and rank k in the case where k is a sum of consecutive powers of 2. Moreover, we present new constructions for 1-subset-regular self-complementary uniform hypergraphs, and prove that these necessary conditions are sufficient for all k, in the case where t = 1.https://onlinelibrary.wiley.com/doi/abs/10.1002/jcd.2028
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
Erdos-Hajnal-type theorems in hypergraphs
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free,
that is, it does not contain an induced copy of a given graph H, then it must
contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0
depends only on the graph H. Except for a few special cases, this conjecture
remains wide open. However, it is known that a H-free graph must contain a
complete or empty bipartite graph with parts of polynomial size. We prove an
analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform
hypergraph on n vertices is H-free, for any given H, then it must contain a
complete or empty tripartite subgraph with parts of order c(log n)^{1/2 +
d(H)}, where d(H) > 0 depends only on H. This improves on the bound of c(log
n)^{1/2}, which holds in all 3-uniform hypergraphs, and, up to the value of the
constant d(H), is best possible. We also prove that, for k > 3, no analogue of
the standard Erdos-Hajnal conjecture can hold in k-uniform hypergraphs. That
is, there are k-uniform hypergraphs H and sequences of H-free hypergraphs which
do not contain cliques or independent sets of size appreciably larger than one
would normally expect.Comment: 15 page
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
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