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Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.
Books versus Triangles
, 2009
"... A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “line ..."
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A book of size b in a graph is an edge that lies in b triangles. Consider a graph G with n vertices and ⌊n 2 /4 ⌋ + 1 edges. Rademacher proved that G contains at least ⌊n/2 ⌋ triangles, and Erdős conjectured and Edwards proved that G contains a book of size at least n/6. We prove the following “linear combination ” of these two results. Suppose that α ∈ (1/2, 1) and the maximum size of a book in G is less than αn/2. Then G contains at least α(1 − α) n2 4 − o(n2) triangles as n → ∞. This is asymptotically sharp. On the other hand, for every α ∈ (1/3, 1/2), there exists β> 0 such that G contains at least βn 3 triangles. It remains an open problem to determine the largest possible β in terms of α. Our proof uses the RuzsaSzemerédi theorem. 1
On the number of K4saturating edges
, 2014
"... Let G be a K4free graph, an edge in its complement is a K4saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any nvertex K4free graph G with bn2/4c + 1 edges, one can find at least (1 + o(1))n ..."
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Let G be a K4free graph, an edge in its complement is a K4saturating edge if the addition of this edge to G creates a copy of K4. Erdős and Tuza conjectured that for any nvertex K4free graph G with bn2/4c + 1 edges, one can find at least (1 + o(1))n