Some results and problems on clique coverings of hypergraphs

For a $k$-uniform hypergraph $F$ we consider the parameter $\Theta(F)$, the minimum size of a clique cover of the of $F$. We derive bounds on $\Theta(F)$ for $F$ belonging to various classes of hypergraphs.Comment: 14 page

Counting small cliques in 3-uniform hypergraphs

Many applications of Szemerédi's Regularity Lemma for graphs are based on the following counting result. If ${\mathcal G}$ is an $s$-partite graph with partition $V({\mathcal G}) =\bigcup_{i=1}^{s} V_i$, $\vert V_i\vert =m$ for all $i\in [s]$, and all pairs $(V_i, V_j)$, $1\leq i < j\leq s$, are $\epsilon$-regular of density $d$, then $\mathcal{G}$ contains $(1\pm f(\epsilon))d^{({s\atop 2})}m^s$ cliques $K_{s}$, provided $\epsilon<\epsilon(d)$, where $f(\epsilon)$ tends to 0 as $\epsilon$ tends to 0. Guided by the regularity lemma for 3-uniform hypergraphs established earlier by Frankl and Rödl, Nagle and Rödl proved a corresponding counting lemma. Their proof is rather technical, mostly due to the fact that the ‘quasi-random’ hypergraph arising after application of Frankl and Rödl's regularity lemma is ‘sparse’, and consequently difficult to handle. When the ‘quasi-random’ hypergraph is ‘dense’ Kohayakawa, Rödl and Skokan (J. Combin. Theory Ser. A 97 307–352) found a simpler proof of the counting lemma. Their result applies even to $k$-uniform hypergraphs for arbitrary $k$. While the Frankl–Rödl regularity lemma will not render the dense case, in this paper, for $k=3$, we are nevertheless able to reduce the harder, sparse case to the dense case. Namely, we prove that a ‘dense substructure’ randomly chosen from the ‘sparse $\delta$-regular structure’ is $\delta$-regular as well. This allows us to count the number of cliques (and other subhypergraphs) using the Kohayakawa–Rödl–Skokan result, and provides an alternative proof of the counting lemma in the sparse case. Since the counting lemma in the dense case applies to $k$-uniform hypergraphs for arbitrary $k$, there is a possibility that the approach of this paper can be adopted to the general case as well

Equivalent conditions for regularity (extended abstract)

) Y. Kohayakawa 1? , V. Rodl 2 , and J. Skokan 2 1 Instituto de Matem&apos;atica e Estat&apos;istica, Universidade de S~ao Paulo, Rua do Mat~ao 1010, 05508--900 S~ao Paulo, Brazil [email protected] 2 Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA frodl,[email protected] Abstract. Haviland and Thomason and Chung and Graham were the first to investigate systematically some properties of quasi-random hypergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1=2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation of a hypergraph. Chung and Graham proved that having small deviation is equivalent to a variety of other properties that describe quasi-randomness. In this note, we consider the concept of discrepancy for k-uniform hypergraphs with an arbitrary constant density d (0 ! d ! ..

On the Size of Set Systems on [n] Not Containing Weak (r, ∆)-Systems

Let r 3 be an integer. A weak (r; ∆)-system is a family of r sets such that all pairwise intersections among the members have the same cardinality. We show that for n large enough, there exists a family F of subsets of [n] such that F does not contain a weak (r; ∆)-system and jF j 2 1 3 \Deltan 1=5 log 4=5 (r\Gamma1) : This improves an earlier result of P. Erdős and E. Szemerédi [ES 78] (cf. [E 90])

Extremal Problems on Set Systems

For a family F (k) = fF 2 ; : : : ; F t g of k-uniform hypergraphs let ex(n; F (k)) denote the maximum number of k-tuples which a k-uniform hypergraph on n vertices may have, while not containing any member of F (k). Let rk (n) denote the maximum cardinality of a set of integers Z [n], where Z contains no arithmetic progression of length k

Families of triples with high minimum degree are hamiltonian

In this paper we show that every family of triples, that is, a 3-uniform hypergraph, with minimum degree at least contains a tight Hamiltonian cycl

REGULAR PAIRS IN SPARSE RANDOM GRAPHS I

We consider bipartite subgraphs of sparse random graphs that are regular in the sense of Szemerédi and, among other things, show that they must satisfy a certain local pseudorandom property. This property and its consequences turn out to be useful when considering embedding problems in subgraphs of sparse random graphs