# Erdős–Stone theorem

In extremal graph theory, the Erdős–Stone theorem is an asymptotic result generalising Turán's theorem to bound the number of edges in an H-free graph for a non-complete graph H. It is named after Paul Erdős and Arthur Stone, who proved it in 1946, and it has been described as the “fundamental theorem of extremal graph theory”.

## Extremal functions of Turán graphs

The extremal function ex(nH) is defined to be the maximum number of edges in a graph of order n not containing a subgraph isomorphic to H. Turán's theorem says that ex(nKr) = tr − 1(n), the order of the Turán graph, and that the Turán graph is the unique extremal graph. The Erdős–Stone theorem extends this to graphs not containing Kr(t), the complete r-partite graph with t vertices in each class (equivalently the Turán graph T(rt,r)):

${\mbox{ex}}(n;K_{r}(t))=\left({\frac {r-2}{r-1}}+o(1)\right){n \choose 2}.$ ## Extremal functions of arbitrary non-bipartite graphs

If H is an arbitrary graph whose chromatic number is r > 2, then H is contained in Kr(t) whenever t is at least as large as the largest color class in an r-coloring of H, but it is not contained in the Turán graph T(n,r − 1) (because every subgraph of this Turán graph may be colored with r − 1 colors). It follows that the extremal function for H is at least as large as the number of edges in T(n,r − 1), and at most equal to the extremal function for Kr(t); that is,

${\mbox{ex}}(n;H)=\left({\frac {r-2}{r-1}}+o(1)\right){n \choose 2}.$ For bipartite graphs H, however, the theorem does not give a tight bound on the extremal function. It is known that, when H is bipartite, ex(nH) = o(n2), and for general bipartite graphs little more is known. See Zarankiewicz problem for more on the extremal functions of bipartite graphs.

## Quantitative results

Several versions of the theorem have been proved that more precisely characterise the relation of n, r, t and the o(1) term. Define the notation sr(n) (for 0 < ε < 1/(2(r − 1))) to be the greatest t such that every graph of order n and size

$\left({\frac {r-2}{2(r-1)}}+\varepsilon \right)n^{2}$ contains a Kr(t).

Erdős and Stone proved that

$s_{r,\varepsilon }(n)\geq \left(\underbrace {\log \cdots \log } _{r-1}n\right)^{1/2}$ for n sufficiently large. The correct order of sr(n) in terms of n was found by Bollobás and Erdős: for any given r and ε there are constants c1(r, ε) and c2(r, ε) such that c1(r, ε) log n < sr(n) < c2(r, ε) log n. Chvátal and Szemerédi then determined the nature of the dependence on r and ε, up to a constant:

${\frac {1}{500\log(1/\varepsilon )}}\log n for sufficiently large n.