The de Bruijn-Erdos Theorem for hypergraphs

2011

More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a basic instance of the hypergraph Tur\'an problem (with a T-edge matching as the excluded hypergraph), progress on this question has remained elusive. In this paper, we verify this conjecture for all T < N/(3K^2). This improves upon the best previously known range T = O(N/K^3), which dates back to the 1970's.

Annals of Operations Research, 2018

One of the primary objectives of this paper is to generalize Hunter's bound to m-regular hypergraphs. In particular, new upper and lower bounds for the probability of the union of events is provided in this paper. The lower bounds generalize Hunter's bound as well. All new bounds are derived from the dual feasible solutions of Boole's linear programming problem.

2020

This work is motivated by the open conjecture concerning the size of a minimum vertex cover in a partitioned hypergraph. In an r-uniform r-partite hypergraph, the size of the minimum vertex cover C is conjectured to be related to the size of its maximum matching M by the relation (|C| ≤ (r − 1)|M |). In fact it is not known whether this conjecture holds when |M | = 1. We consider r-partite hypergraphs with maximal matching size |M | = 1, and pose a novel algorithmic approach to finding a vertex cover of size (r − 1) in this case. We define a reactive hypergraph to be a back-and-forth algorithm for a hypergraph which chooses new edges in response to a choice of vertex cover, and prove that this algorithm terminates for all hypergraphs of orders r = 3 and 4. We introduce the idea of optimizing the size of the reactive hypergraph and find that the reactive hypergraph terminates for r = 5...20. We then consider the case where the intersection of any two edges is exactly 1. We prove bounds on the size of this 1-intersecting hypergraph and relate the 1-intersecting hypergraph maximization problem to mutually orthogonal Latin squares. We propose a generative algorithm for 1-intersecting hypergraphs of maximal size for prime powers r − 1 = p d under the constraint pd + 1 is also a prime power of the same form, and therefore pose a new generating algorithm for MOLS based upon intersecting hypergraphs. We prove this algorithm generates a valid set of mutually orthogonal Latin squares and prove the construction guarantees certain symmetric properties. We conclude that a conjecture by Lovász [1], that the inequality in Ryser's Conjecture cannot be improved when (r − 1) is a prime power, is correct for the 1-intersecting hypergraph of prime power orders.

Israel Journal of Mathematics, 2006

We present alternative proofs of density versions of some combinatorial partition theorems originally obtained by E. Szemerédi, H. Furstenberg, and Y. Katznelson.

Electronic Notes in Discrete Mathematics, 2010

We extend the Erdős-Gallai Theorem for Berge paths in r-uniform hypergraphs. We also find the extremal hypergraphs avoiding t-tight paths of a given length and consider this extremal problem for other definitions of paths in hypergraphs.

Mathematical Programming, 2003

A result of Balas and Yu (1989) states that the number of maximal independent sets of a graph G is at most δ p + 1, where δ is the number of pairs of vertices in G at distance 2, and p is the cardinality of a maximum induced matching in G. In this paper, we give an analogue of this result for hypergraphs and, more generally, for subsets of vectors B in the product of n lattices L = L1 × • • • × Ln, where the notion of an induced matching in G is replaced by a certain binary tree each internal node of which is mapped into B. We show that our bounds may be nearly sharp for arbitrarily large hypergraphs and lattices. As an application, we prove that the number of maximal infeasible vectors x ∈ L = L1 × • • • × Ln for a system of polymatroid inequalities f1(x) ≥ t1,. .. , fr(x) ≥ tr does not exceed max{Q, β log t/c(2Q,β) }, where β is the number of minimal feasible vectors for the system, Q = |L1| +. .. + |Ln|, t = max{t1,. .. , tr}, and c(ρ, β) is the unique positive root of the equation 2 c (ρ c/ log β − 1) = 1. This bound is nearly sharp for the Boolean case L = {0, 1} n , and it allows for the efficient generation of all minimal feasible sets to a given system of polymatroid inequalities with quasi-polynomially bounded right-hand sides t1,. .. , tm.

2017

Trees fill many extremal roles in graph theory, being minimally connected and serving a critical role in the definition of $n$-good graphs. In this article, we consider the generalization of trees to the setting of $r$-uniform hypergraphs and how one may extend the notion of $n$-good graphs to this setting. We prove numerous bounds for $r$-uniform hypergraph Ramsey numbers involving trees and complete hypergraphs and show that in the $3$-uniform case, all trees are $n$-good when $n$ is odd or $n$ falls into specified even cases.

2010

The competition hypergraph CH(D) of a digraph D is the hypergraph such that the vertex set is the same as D and e ⊆ V (D) is a hyperedge if and only if e contains at least 2 vertices and e coincides with the in-neighborhood of some vertex v in the digraph D. Any hypergraph H with sufficiently many isolated vertices is the competition hypergraph of an acyclic digraph. The hypercompetition number hk(H) of a hypergraph H is defined to be the smallest number of such isolated vertices.

European Journal of Combinatorics, 2012

For fixed positive integers r, k and ℓ with 1 ≤ ℓ < r and an r-uniform hypergraph H, let κ(H, k, ℓ) denote the number of k-colorings of the set of hyperedges of H for which any two hyperedges in the same color class intersect in at least ℓ elements. Consider the function KC(n, r, k, ℓ) = maxH∈H n κ (H, k, ℓ), where the maximum runs over the family Hn of all r-uniform hypergraphs on n vertices. In this paper, we determine the asymptotic behavior of the function KC(n, r, k, ℓ) for every fixed r, k and ℓ and describe the extremal hypergraphs. This variant of a problem of Erdős and Rothschild, who considered edge colorings of graphs without a monochromatic triangle, is related to the Erdős-Ko-Rado Theorem on intersecting systems of sets [Intersection Theorems for Systems of Finite Sets, Quarterly Journal of Mathematics, Oxford Series, Series 2, 12 (1961), 313-320].

Discrete Mathematics

In this note we asymptotically determine the maximum number of hyperedges possible in an r-uniform, connected n-vertex hypergraph without a Berge path of length k, as n and k tend to infinity. We show that, unlike in the graph case, the multiplicative constant is smaller with the assumption of connectivity.

arXiv: Combinatorics, 2018

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erdős-Ko-Rado Theorem, and generalizes earlier results of C...

Discrete Mathematics, 2019

Let fr(n) represent the minimum number of complete r-partite rgraphs required to partition the edge set of the complete r-uniform hypergraph on n vertices. The Graham-Pollak theorem states that f2(n) = n − 1. An upper bound of (1+o(1)) n ⌊ r 2 ⌋ was known. Recently this was improved to 14 15 (1 + o(1)) n ⌊ r 2 ⌋ for even r ≥ 4. A bound of r 2 (14 15) r 4 + o(1) (1 + o(1)) n ⌊ r 2 ⌋ was also proved recently. The smallest odd r for which cr < 1 that was known was for r = 295. In this note we improve this to c113 < 1 and also give better upper bounds for fr(n), for small values of even r.

Discrete Mathematics, 2008

Consider a hypergraph of rank r > 2 with m edges, independence number and edge cover number . We prove the inequality

European Journal of Combinatorics, 1984

Let F(n) denote the maximum number of distinct subsets of an n-element set such that there are no four distinct subsets: A, B, C, D with A v B = C v D. We prove that 2<n-Ios 3 ll 3 -2.;:; F( n).;:; 2< 3 n+Z)/ 4 • We use probability theory for the proof of both the lower and upper bounds. Some related problems are considered, too.

SIAM Journal on Discrete Mathematics, 2018

Let G be a finite abelian group, and r be a multiple of its exponent. The generalized Erdős-Ginzburg-Ziv constant sr(G) is the smallest integer s such that every sequence of length s over G has a zero-sum subsequence of length r. We show that s 2m (Z d 2) ≤ Cm2 d/m + O(1) when d → ∞, and s 2m (Z d 2) ≥ 2 d/m + 2m − 1 when d = km. We use results on sr(G) to prove new bounds for the codegree Turán density of complete r-graphs.