Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs

General upper tail estimates are given for counting edges in a random induced subhypergraph of a fixed hypergraph H, with an easy proof by estimating the moments. As an application we consider the numbers of arithmetic progressions and Schur triples in random subsets of integers. In the second part of the paper we return to the subgraph counts in random graphs and provide upper tail estimates in the rooted case.Comment: 15 page

Clique Factors: Extremal and Probabilistic Perspectives

A K_r-factor in a graph G is a collection of vertex-disjoint copies of K_r covering the vertex set of G. In this thesis, we investigate these fundamental objects in three settings that lie at the intersection of extremal and probabilistic combinatorics. Firstly, we explore pseudorandom graphs. An n-vertex graph is said to be (p,β)-bijumbled if for any vertex sets A, B ⊆ V (G), we have e( A, B) = p| A||B| ± β√|A||B|. We prove that for any 3 ≤ r ∈ N and c > 0 there exists an ε > 0 such that any n-vertex (p, β)-bijumbled graph with n ∈ rN, δ(G) ≥ c p n and β ≤ ε p^{r −1} n, contains a K_r -factor. This implies a corresponding result for the stronger pseudorandom notion of (n, d, λ)-graphs. For the case of K_3-factors, this result resolves a conjecture of Krivelevich, Sudakov and Szabó from 2004 and it is tight due to a pseudorandom triangle-free construction of Alon. In fact, in this case even more is true: as a corollary to this result, we can conclude that the same condition of β = o( p^2n) actually guarantees that a (p, β)-bijumbled graph G contains every graph on n vertices with maximum degree at most 2. Secondly, we explore the notion of robustness for K_3-factors. For a graph G and p ∈ [0, 1], we denote by G_p the random sparsification of G obtained by keeping each edge of G independently, with probability p. We show that there exists a C > 0 such that if p ≥ C (log n)^{1/3}n^{−2/3} and G is an n-vertex graph with n ∈ 3N and δ(G) ≥ 2n/3 , then with high probability G_p contains a K_3-factor. Both the minimum degree condition and the probability condition, up to the choice of C, are tight. Our result can be viewed as a common strengthening of the classical extremal theorem of Corrádi and Hajnal, corresponding to p = 1 in our result, and the famous probabilistic theorem of Johansson, Kahn and Vu establishing the threshold for the appearance of K_3-factors (and indeed all K_r -factors) in G (n, p), corresponding to G = K_n in our result. It also implies a first lower bound on the number of K_3-factors in graphs with minimum degree at least 2n/3, which gets close to the truth. Lastly, we consider the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze and Martin, where one starts with a dense graph and then adds random edges to it. Specifically, given any fixed 0 < α < 1 − 1/r we determine how many random edges one must add to an n-vertex graph G with δ(G) ≥ α n to ensure that, with high probability, the resulting graph contains a K_r -factor. As one increases α we demonstrate that the number of random edges required ‘jumps’ at regular intervals, and within these intervals our result is best-possible. This work therefore bridges the gap between the seminal work of Johansson, Kahn and Vu mentioned above, which resolves the purely random case, i.e., α = 0, and that of Hajnal and Szemerédi (and Corrádi and Hajnal for r = 3) showing that when α ≥ 1 − 1/r the initial graph already hosts the desired K_r -factor.Ein K_r -Faktor in einem Graphen G ist eine Sammlung von Knoten-disjunkten Kopien von K_r , die die Knotenmenge von G überdecken. Wir untersuchen diese Objekte in drei Kontexten, die an der Schnittstelle zwischen extremaler und probabilistischer Kombinatorik liegen. Zuerst untersuchen wir Pseudozufallsgraphen. Ein Graph heißt (p,β)-bijumbled, wenn für beliebige Knotenmengen A, B ⊆ V (G) gilt e( A, B) = p| A||B| ± β√|A||B|. Wir beweisen, dass es für jedes 3 ≤ r ∈ N und c > 0 ein ε > 0 gibt, so dass jeder n-Knoten (p, β)-bijumbled Graph mit n ∈ rN, δ(G) ≥ c p n und β ≤ ε p^{r −1} n, einen K_r -Faktor enthält. Dies impliziert ein entsprechendes Ergebnis für den stärkeren Pseudozufallsbegriff von (n, d, λ)-Graphen. Im Fall von K_3-Faktoren, löst dieses Ergebnis eine Vermutung von Krivelevich, Sudakov und Szabó aus dem Jahr 2004 und ist durch eine pseudozufällige K_3-freie Konstruktion von Alon bestmöglich. Tatsächlich ist in diesem Fall noch mehr wahr: als Korollar dieses Ergebnisses können wir schließen, dass die gleiche Bedingung von β = o( p^2n) garantiert, dass ein (p, β)-bijumbled Graph G jeden Graphen mit maximalem Grad 2 enthält. Zweitens untersuchen wir den Begriff der Robustheit für K_3-Faktoren. Für einen Graphen G und p ∈ [0, 1] bezeichnen wir mit G_p die zufällige Sparsifizierung von G, die man erhält, indem man jede Kante von G unabhängig von den anderen Kanten mit einer Wahrscheinlichkeit p behält. Wir zeigen, dass, wenn p ≥ C (log n)^{1/3}n^{−2/3} und G ein n-Knoten-Graph mit n ∈ 3N und δ(G) ≥ 2n/3 ist, G_pmit hoher Wahrscheinlichkeit (mhW) einen K_3-Faktor enthält. Sowohl die Bedingung des minimalen Grades als auch die Wahrscheinlichkeitsbedingung sind bestmöglich. Unser Ergebnis ist eine Verstärkung des klassischen extremalen Satzes von Corrádi und Hajnal, entsprechend p = 1 in unserem Ergebnis, und des berühmten probabilistischen Satzes von Johansson, Kahn und Vu, der den Schwellenwert für das Auftreten eines K_3-Faktors (und aller K_r -Faktoren) in G (n, p) festlegt, entsprechend G = K_n in unserem Ergebnis. Es impliziert auch eine erste untere Schranke für die Anzahl der K_3-Faktoren in Graphen mit einem minimalen Grad von mindestens 2n/3, die der Wahrheit nahe kommt. Schließlich betrachten wir die Situation von zufällig gestörten Graphen; ein Modell, bei dem man mit einem dichten Graphen beginnt und dann zufällige Kanten hinzufügt. Wir bestimmen, bei gegebenem 0 < α < 1 − 1/r, wie viele zufällige Kanten man zu einem n-Knoten-Graphen G mit δ(G) ≥ α n hinzufügen muss, um sicherzustellen, dass der resultierende Graph mhW einen K_r -Faktor enthält. Wir zeigen, dass, wenn man α erhöht, die Anzahl der benötigten Zufallskanten in regelmäßigen Abständen “springt", und innerhalb dieser Abstände unser Ergebnis bestmöglich ist. Diese Arbeit schließt somit die Lücke zwischen der oben erwähnten bahnbrechenden Arbeit von Johansson, Kahn und Vu, die den rein zufälligen Fall, d.h. α = 0, löst, und der Arbeit von Hajnal und Szemerédi (und Corrádi und Hajnal für r = 3), die zeigt, dass der ursprüngliche Graph bereits den gewünschten K_r -Faktor enthält, wenn α ≥ 1 − 1/r ist

An analytic approach to sparse hypergraphs: hypergraph removal

An analytic approach to sparse hypergraphs: hypergraph removal, Discrete Analysis 2018:3, 47 pp. The famous triangle removal lemma of Ruzsa and Szemerédi states that for every $\epsilon>0$ there exists $\delta>0$ such that every graph $G$ on $n$ vertices with at most $\delta n^3$ triangles has a triangle-free subgraph $G'$ such $|E(G)\setminus E(G')|\leq\epsilon n^2$. To put it less formally, a graph that is almost triangle free can be approximated by a graph that is exactly triangle free. A surprising application of this innocent looking result is a proof of Roth's theorem, that for every $\delta>0$ there exists $n$ such that every subset $A\subset\{1,2,\dots,n\}$ of size at least $\delta n$ contains an arithmetic progression of length 3. To prove this deduction, one defines a tripartite graph with appropriate vertex sets $X,Y,Z$ (one can, for instance, take each set to be $\{1,2,\dots,3n\}$), with $xy$ an edge if $y-x\in A$, $yz$ an edge if $z-y\in A$, and $xz$ an edge if $(z-x)/2\in A$. Then a triangle $xyz$ in the graph corresponds to an arithmetic progression $y-x,z-x,z-y$ in $A$ (which may be degenerate, but there cannot be too many degenerate ones). This lemma has been generalized in several different directions. One is to a "simplex removal lemma" for hypergraphs. For 3-uniform hypergraphs, for example, it states that for every $\epsilon>0$ there exists $\delta>0$ such that if a 3-uniform hypergraph $H$ on $n$ vertices contains at most $\delta n^4$ simplices (that is, configurations of the form $xyz,xyw,xzw,yzw$), then one can throw away at most $\epsilon n^3$ triples from $H$ in order to obtain a subhypergraph $H'$ that contains no simplices at all. Another direction of generalization is to a "sparse random version". Here the idea is to start with a random graph $U$ with $n$ vertices and edge probability $n^{-\alpha}$ for some suitable $\alpha>0$, and to prove a result about subgraphs $G$ of $U$, where the various densities are measured relative to $U$. Thus, $G$ is deemed to have few triangles if the number of triangles in $G$ is at most $\delta$ times the number of triangles in $U$, and $G'$ is deemed to be a good approximation of $G$ if it differs from $G$ by at most $\epsilon$ times the number of edges in $U$. Of course, one can try to generalize the result in both directions simultaneously in order to obtain a sparse random version of the simplex removal lemma. The purpose of this paper is to give a new way of proving such a generalization. The methods used are infinitary, continuing a line of research initiated by Lovász and various coauthors with their introduction of graph and hypergraph limits. The novelty of the paper is that it manages to apply an infinitary approach of this kind in the sparse random context, which was not obviously possible and which required the author to work out how to deal with some serious-looking obstacles