7 research outputs found
Quasirandomness in hypergraphs
An -vertex graph of edge density is considered to be quasirandom
if it shares several important properties with the random graph . A
well-known theorem of Chung, Graham and Wilson states that many such `typical'
properties are asymptotically equivalent and, thus, a graph possessing one
such property automatically satisfies the others.
In recent years, work in this area has focused on uncovering more quasirandom
graph properties and on extending the known results to other discrete
structures. In the context of hypergraphs, however, one may consider several
different notions of quasirandomness. A complete description of these notions
has been provided recently by Towsner, who proved several central equivalences
using an analytic framework. We give short and purely combinatorial proofs of
the main equivalences in Towsner's result.Comment: 19 page
Small rainbow cliques in randomly perturbed dense graphs
For two graphs G and H, write G
rbw
−→ H if G has the property that every proper colouring
of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey
properties in randomly perturbed dense graphs, which are unions of the form G ∪ G(n, p), where
G is an n-vertex graph with edge-density at least d > 0, and d is independent of n.
In a companion paper, we proved that the threshold for the property G ∪ G(n, p)
rbw
−→ K` is
n
−1/m2(Kd`/2e)
, whenever ` ≥ 9. For smaller `, the thresholds behave more erratically, and for
4 ≤ ` ≤ 7 they deviate downwards significantly from the aforementioned aesthetic form capturing
the thresholds for large cliques.
In particular, we show that the thresholds for ` ∈ {4, 5, 7} are n
−5/4
, n
−1
, and n
−7/15, respectively. For ` ∈ {6, 8} we determine the threshold up to a (1 + o(1))-factor in the exponent: they
are n
−(2/3+o(1)) and n
−(2/5+o(1)), respectively. For ` = 3, the threshold is n
−2
; this follows from
a more general result about odd cycles in our companion paper
Large rainbow cliques in randomly perturbed dense graphs
For two graphs G and H, write G⟶ rbw H if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪ G(n, p), where G is an n-vertex graph with edge-density at least d, and d is a constant that does not depend on n. We determine the threshold for the property G∪ G(n, p) ⟶ rbw Ks for every s. We show that for s≥ 9 the threshold is n-1/m2(K⌈s/2⌉) ; in fact, our 1-statement is a supersaturation result. This turns out to (almost) be the threshold for s= 8 as well, but for every 4 ≤ s≤ 7, the threshold is lower and is different for each 4 ≤ s≤ 7. Moreover, we prove that for every ℓ≥ 2 the threshold for the property G∪ G(n, p) ⟶ rbw C2â„“-1 is n- 2 ; in particular, the threshold does not depend on the length of the cycle C2â„“-1. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all
Rainbow Cliques in Randomly Perturbed Dense Graphs
For two graphs G and H, write G⟶ rbw H if G has the property that every proper colouring of its edges yields a rainbow copy of H. We study the thresholds for such so-called anti-Ramsey properties in randomly perturbed dense graphs, which are unions of the form G∪ G(n, p), where G is an n-vertex graph with edge-density at least d, and d is a constant that does not depend on n. We determine the threshold for the property G∪ G(n, p) ⟶ rbw Ks for every s. We show that for s≥ 9 the threshold is n-1/m2(K⌈s/2⌉) ; in fact, our 1-statement is a supersaturation result. This turns out to (almost) be the threshold for s= 8 as well, but for every 4 ≤ s≤ 7, the threshold is lower and is different for each 4 ≤ s≤ 7. Moreover, we prove that for every ℓ≥ 2 the threshold for the property G∪ G(n, p) ⟶ rbw C2â„“-1 is n- 2 ; in particular, the threshold does not depend on the length of the cycle C2â„“-1. It is worth mentioning that for even cycles, or more generally for any fixed bipartite graph, no random edges are needed at all
Quasirandomness in hypergraphs
A graph G is called quasirandom if it possesses typical properties of the corresponding random graph G(n,p) with the same edge density as G. A well-known theorem of Chung, Graham and Wilson states that, in fact, many such ‘typical’ properties are asymptotically equivalent and, thus, a graph G possessing one property immediately satisfies the others. In recent years, more quasirandom graph properties have been found and extensions to hypergraphs have been explored. For the latter, however, there exist several distinct notions of quasirandomness. A complete description of these notions has been provided recently by Towsner, who proved several central equivalences using an analytic framework. The purpose of this paper is to give short purely combinatorial proofs of most of Towsner's results