Abstract
Extremal graph theory is the study of graphs which have a critical behaviour with respect to some graph parameter within a certain class characterized by some graph property. The typical example that we will consider in this chapter consists in finding the maximum number of edges that a graph can have within the class of graphs which do not contain a fixed subgraph H. The main result in this area is the Erdős–Stone theorem. This theorem provides an asymptotic expression for the maximum number of edges a graph can have which has no subgraph H. This expression depends only on the chromatic number of the graph H. The theorem is not informative when H is bipartite, and the last part of the chapter is devoted to study this case in which finite geometries will reappear.
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References
P. Erdős, A.H. Stone, On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091 (1946)
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Ball, S., Serra, O. (2024). Extremal Graph Theory. In: A Course in Combinatorics and Graphs. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-55384-4_9
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DOI: https://doi.org/10.1007/978-3-031-55384-4_9
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