Abstract
AbstractThe Turán number ex(n, H) of a graph H is the maximal number of edges in an H-free graph on n vertices. In 1983, Chung and Erdős asked which graphs H with e edges minimise ex(n, H). They resolved this question asymptotically for most of the range of e and asked to complete the picture. In this paper, we answer their question by resolving all remaining cases. Our result translates directly to the setting of universality, a well-studied notion of finding graphs which contain every graph belonging to a certain family. In this setting, we extend previous work done by Babai, Chung, Erdős, Graham and Spencer, and by Alon and Asodi.
Highlights
The following question of Turán dating back to 1941 [27] is one of the most classical problems of graph theory
Given a fixed graph H, what is the maximal number of edges one can have in an n vertex graph which does not contain a copy of H as a subgraph? The answer to this question, denoted ex(n, H), is called the Turán number of H
Turán’s problem leads to another very natural extremal question—what is the largest size of a graph which we cannot avoid as a subgraph in any graph on n vertices and e edges? In other words, what kind of a graph H with a fixed number of edges has minimal Turán number?
Summary
The following question of Turán dating back to 1941 [27] is one of the most classical problems of graph theory. We revisit the original question of Chung and Erdos They say a graph H is (n, e)unavoidable if every graph on n vertices and e edges contains a copy of H as a subgraph. It is worth noting that the unavoidable graph we use in order to obtain the lower bound in the second part of the above theorem is the random Erdos–Renyi graph. This is in stark contrast to the very structured graph used by Chung and Erdos in regimes of (ii)a and (iii) above. For part of the regime both of these very different examples are extremal, up to a constant factor
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