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Open problems of Paul Erdős in graph theory
 J. GRAPH THEORY
, 1997
"... The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory a ..."
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The main treasure that Paul Erdős has left us is his collection of problems, most of which are still open today. These problems are seeds that Paul sowed and watered by giving numerous talks at meetings big and small, near and far. In the past, his problems have spawned many areas in graph theory and beyond (e.g., in number theory, probability, geometry, algorithms and complexity theory). Solutions or partial solutions to Erdős problems usually lead to further questions, often in new directions. These problems provide inspiration and serve as a common focus for all graph theorists. Through the problems, the legacy of Paul Erdős continues (particularly if solving one of these problems results in creating three new problems, for example.) There is a huge literature of almost 1500 papers written by Erdős and his (more than 460) collaborators. Paul wrote many problem papers, some of which appeared in various (really hardtofind) proceedings. Here is an attempt to collect and organize these problems in the area of graph theory. The list here is by no means complete or exhaustive. Our goal is to state the problems, locate the sources, and provide the references related to these problems. We will include the earliest and latest known references without covering the entire history of the problems because of space limitations. (The most uptodate list of Erdős’ papers can be found in [65]; an electronic file is maintained by Jerry Grossman at
Topological Graphs with no SelfIntersecting Cycle of Length 4
 CONTEMPORARY MATHEMATICS
"... Let G be a topological graph on n vertices in the plane, i.e., a graph drawn in the plane with its vertices represented as points and its edges represented as Jordan arcs connecting pairs of points. It is shown that if no two edges of any cycle of length 4 in G cross an odd number of times, then E ..."
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Let G be a topological graph on n vertices in the plane, i.e., a graph drawn in the plane with its vertices represented as points and its edges represented as Jordan arcs connecting pairs of points. It is shown that if no two edges of any cycle of length 4 in G cross an odd number of times, then E(G)  = O(n 8/5).
Subgraph densities in signed graphons and the local SimonovitsSidorenko conjecture
 ELECTRON J. COMBIN
, 2011
"... We prove inequalities between the densities of various bipartite subgraphs in signed graphs. One of the main inequalities is that the density of any bipartite graph with girth 2r cannot exceed the density of the 2rcycle. This study is motivated by the Simonovits–Sidorenko conjecture, which states t ..."
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We prove inequalities between the densities of various bipartite subgraphs in signed graphs. One of the main inequalities is that the density of any bipartite graph with girth 2r cannot exceed the density of the 2rcycle. This study is motivated by the Simonovits–Sidorenko conjecture, which states that the density of a bipartite graph F with m edges in any graph G is at least the mth power of the edge density of G. Another way of stating this is that the graph G with given edge density minimizing the number of copies of F is, asymptotically, a random graph. We prove that this is true locally, i.e., for graphs G that are “close” to a random graph. Both kinds of results are treated in the framework of graphons (2variable functions serving as limit objects for graph sequences), which in this context was already
Recent developments in graph Ramsey theory
"... Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any twocolouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been ..."
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Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any twocolouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.
NonThreeColourable Common Graphs Exist
, 2012
"... A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectu ..."
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A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Stovíček and Thomason from 1996 we show that the 5wheel is common. This provides the first example of a common graph that is not threecolourable.
A survey of Turán problems for expansions
, 2015
"... The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n ..."
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The rexpansion G+ of a graph G is the runiform hypergraph obtained from G by enlarging each edge of G with a vertex subset of size r − 2 disjoint from V (G) such that distinct edges are enlarged by disjoint subsets. Let exr(n, F) denote the maximum number of edges in an runiform hypergraph with n vertices not containing any copy of the runiform hypergraph F. Many problems in extremal set theory ask for the determination of exr(n,G +) for various graphs G. We survey these Turántype problems, focusing on recent developments. 1
Two Approaches to Sidorenko’s Conjecture
"... Sidorenko’s conjecture states that for every bipartite graphH on {1, · · · , k} ∫ ∏ ..."
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Sidorenko’s conjecture states that for every bipartite graphH on {1, · · · , k} ∫ ∏