From social networks such as Facebook, the World Wide Web and the Internet, to the complex interactions between proteins in the cells of our bodies, we constantly face the challenge of understanding the structure and development of networks. The theory of random graphs provides a framework for this understanding, and in this book the authors give a gentle introduction to the basic tools for understanding and applying the theory. Part I includes sufficient material, including exercises, for a one semester course at the advanced undergraduate or beginning graduate level. The reader is then well prepared for the more advanced topics in Parts II and III. A final part provides a quick introduction to the background material needed. All those interested in discrete mathematics, computer science or applied probability and their applications will find this an ideal introduction to the subject.

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Introduction to random graphs

Many important theorems and conjectures in combinatorics, such as the the- orem of Szemeredi on arithmetic progressions and the Erd˝os-Stone Theorem in extremal graph theory, can be phrased as statements about families of independent sets in certain uniform hypergraphs. In recent years, an important trend in the area has been to extend such classical results to the so-called 'sparse random setting'. This line of research has recently culminated in the breakthroughs of Conlon and Gowers and of Schacht, who de- veloped general tools for solving problems of this type. Although these two papers solved very similar sets of longstanding open problems, the methods used are very different from one another and have different strengths and weaknesses. In this paper, we provide a third, completely different approach to proving extremal and structural results in sparse random sets that also yields their natural 'counting' coun- terparts. We give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets. We then derive many interesting results as fairly straightforward consequences of this abstract theorem. In particular, we prove the well- known conjecture of Kohayakawa, Luczak, and Rodl, a probabilistic embedding lemma for sparse graphs, for all 2-balanced graphs. We also give alternative proofs of many of the results of Conlon and Gowers and Schacht, such as sparse random versions of Szemeredi's theorem, the Erd˝os-Stone Theorem and the Erd˝os-Simonovits Stability Theorem, and ob- tain their natural 'counting' versions, which in some cases are considerably stronger. We also obtain new results, such as a sparse version of the Erd˝os-Frankl-Rodl Theorem on the number of H-free graphs and, as a consequence of the K LR conjecture, we extend a re- sult of Rodl and Rucinski on Ramsey properties in sparse random graphs to the general, non-symmetric setting. Similar results have been discovered independently by Saxton and Thomason.

https://www.ams.org/jams/2015-28-03/S0894-0347-2014-00816-X/S0894-0347-2014-00816-X.pdf

Independent sets in hypergraphs

A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This far-reaching result has proved to play a central role in many areas of combinatorics, both ‘pure’ and ‘algorithmic.’ The quest for an equally powerful variant of this lemma for sparse graphs has not yet been successful, but some progress has been achieved recently. The aim of this note is to report on the successes so far.

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Szemerédi's regularity lemma for sparse graphs

Szemeredi's Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in proving theorems for arbitrary graphs whenever the corresponding result is easy for random graphs. In the last few years more and more new results were obtained by using the Regularity Lemma, and Mso some new variants and generalizations appeared. Komlos and Simonovits have written a survey on the topic [96]. The present survey is, in a sense, a continuation of the earlier survey. Here we describe some sample applications and generalizations. To keep the paper self-contained we decided to repeat (sometimes in a shortened form) parts of the first survey, but the emphasis is on new results.

The Regularity Lemma and Its Applications in Graph Theory

We study thresholds for extremal properties of random discrete structures. We determine the threshold for Szemer edi’s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions, and we determine the threshold for Tur an-type problems for random graphs and hypergraphs. In particular, we verify a conjecture of

Extremal results for random discrete structures

For any integer k, we prove the existence of a uniquely k-colourable graph of girth at least g on at most k12(g+1) vertices whose maximal degree is at most 5k13. From this we deduce that, unless NP=RP, no polynomial time algorithm for k-Colourability on graphs G of girth g(G)≥log∣G∣/13logk and maximum degree Δ(G)≤6k13 can exist. We also study several related problems.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0963548398003678

Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth

2 k
 -free subgraph of a random graph may have, obtaining best possible results for a range of p=p(n). Our estimates strengthen previous bounds of Furedi [12] and Haxell, Kohayakawa, and Łuczak [13]. Two main tools are used here: the first one is an upper bound for the number of graphs with large even-girth, i.e., graphs without short even cycles, with a given number of vertices and edges, and satisfying a certain additional pseudorandom condition; the second tool is the powerful result of Ajtai, Komlos, Pintz, Spencer, and Szemeredi [1] on uncrowded hypergraphs as given by Duke, Lefmann, and Rodl [7].

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

We consider the binomial random graphGn,p and determine a sharp threshold function for the edge-Ramsey property Gn,p → (C1 , . . . , Cr ) for all `1, . . . , `r, where C denotes the cycle of length `. As deterministic consequences of our results, we prove the existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above. AMS subject classification code (1991): 05C55, 05C80, 05C38

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Threshold functions for asymmetric Ramsey properties involving cycles

In this note we investigate Ramsey properties of random graphs. The threshold functions for symmetric Ramsey properties with respect to vertex colorings were determined by Łuczak, Rucinski, and Voigt [6]. As a generalization of this problem we consider asymmetric Ramsey properties and establish the values of the threshold functions. Furthermore, we investigate canonical colorings. © 1996 John Wiley & Sons, Inc.

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Threshold functions for asymmetric Ramsey properties with respect to vertex colorings

We form the random poset (n, p) by including each subset of [n]={1,…,n} with probability p and ordering the subsets by inclusion. We investigate the length of the longest chain contained in (n, p). For p≥e/n we obtain the limit distribution of this random variable. For smaller p we give thresholds for the existence of chains which imply that almost surely the length of (n, p) is asymptotically n(log n−log log 1/p)/log 1/p. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 177–194, 2000

Bernd Kreuter

Papers

Uniquely Colourable Graphs and the Hardness of Colouring Graphs of Large Girth

An Extremal Problem For Random Graphs And The Number Of Graphs With Large Even-Girth

Threshold functions for asymmetric Ramsey properties involving cycles

Threshold functions for asymmetric Ramsey properties with respect to vertex colorings

The length of random subsets of Boolean lattices