• Corpus ID: 235358646

Which graphs can be counted in $C_4$-free graphs?

  title={Which graphs can be counted in \$C\_4\$-free graphs?},
  author={David Conlon and Jacob Fox and Benny Sudakov and Yufei Zhao},
For which graphs F is there a sparse F -counting lemma in C4-free graphs? We are interested in identifying graphs F with the property that, roughly speaking, if G is an n-vertex C4free graph with on the order of n edges, then the density of F in G, after a suitable normalization, is approximately at least the density of F in an ǫ-regular approximation of G. In recent work, motivated by applications in extremal and additive combinatorics, we showed that C5 has this property. Here we construct a… 

Tur\'{a}n problems in pseudorandom graphs

The first nontrivial upper bound for this problem when F is the Peterson graph is given and a new construction of the densest known clique-free pseudorandom graphs is given.



The regularity method for graphs with few 4‐cycles

We develop a sparse graph regularity method that applies to graphs with few 4‐cycles, including new counting and removal lemmas for 5‐cycles in such graphs. Some applications include: Every n ‐vertex

Sparse Quasi-Random Graphs

This paper extends the study of quasi-randomness to sparse graphs, i.e., graphs on n vertices with o(n) edges, and it will be shown that many of the quasi- random properties for dense graphs have analogues for sparse graph, while others do not, at least not without additional hypotheses.

On the KŁR conjecture in random graphs

This work proves a variant of the KŁR conjecture which is sufficient for most known applications to random graphs and implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems.


A long-standing problem about the maximal number of edges of a graph not containing a cycle of length 4 is solved and some unsolved problems are mentioned.

On independent sets in hypergraphs

It is proved that if Hn is an n-vertex r+1-uniform hypergraph in which every r-element set is contained in at most d edges, where 0 0 satisfies cr~r/e as ri¾?∞, then cr improves and generalizes several earlier results and gives an application to hypergraph Ramsey numbers involving independent neighborhoods.

Pseudo-random Graphs

Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake,

Szemerédi's regularity lemma for sparse graphs

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

Extremal results in sparse pseudorandom graphs

On Graphs that do not Contain a Thomsen Graph

  • W. G. Brown
  • Mathematics
    Canadian Mathematical Bulletin
  • 1966
A Thomsen graph [2, p. 22] consists of six vertices partitioned into two classes of three each, with every vertex in one class connected to every vertex in the other; it is the graph of the “gas,

Szemerédi's Regularity Lemma for Matrices and Sparse Graphs

  • A. Scott
  • Mathematics, Computer Science
    Combinatorics, Probability and Computing
  • 2011
This paper proves a sparse Regularity Lemma that holds for all graphs and more generally, gives a regularity lemma that holding for arbitrary real matrices.