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Books in graphs
, 2008
"... A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) ..."
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Cited by 1934 (20 self)
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A set of q triangles sharing a common edge is called a book of size q. We write β (n, m) for the the maximal q such that every graph G (n, m) contains a book of size q. In this note 1) we compute β ( n, cn 2) for infinitely many values of c with 1/4 < c < 1/3, 2) we show that if m ≥ (1/4 − α) n 2 with 0 < α < 17 −3 (), and G has no book of size at least graph G1 of order at least
BCC Problem List
, 2000
"... This document contains the problems presented at problem sessions at the British Combinatorial Conference, from BCC12 to the present. All those problems which were published in the Conference proceedings, and some others, are included. I have annotated and updated them with further references wherev ..."
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This document contains the problems presented at problem sessions at the British Combinatorial Conference, from BCC12 to the present. All those problems which were published in the Conference proceedings, and some others, are included. I have annotated and updated them with further references wherever I know of any. Further information on any of the problems are very welcome, and will be included in subsequent versions of the problems. The proposers' addresses are the most recent known to me; again, updates are welcome. From BCC13 onwards, the problems (together with the contributed papers) have been published in the Research Problems section of Discrete Mathematics. Each problem appearing in this section receives a unique number, which I have given in the form DMnnn. 1 BCC12
Book Ramsey Numbers I
, 2008
"... A book Bp is a graph consisting of p triangles sharing a common edge. In this paper we prove that if p ≤ q/6 − o (q) and q is large then the Ramsey number r (Bp, Bq) is given by r (Bp, Bq) = 2q + 3 and the constant 1/6 is essentially best possible. Our proof is based on Szemerédi’s uniformity lemma ..."
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A book Bp is a graph consisting of p triangles sharing a common edge. In this paper we prove that if p ≤ q/6 − o (q) and q is large then the Ramsey number r (Bp, Bq) is given by r (Bp, Bq) = 2q + 3 and the constant 1/6 is essentially best possible. Our proof is based on Szemerédi’s uniformity lemma and a stability result for books.
function, and
, 2001
"... Let g(x; n), with x 2 R+, be a step function for each n. Assuming certain technical hypotheses, we give a constant ® and function f such thatP1 n=1 g(x; n) can be written in the form ®+ P 0<r<x f(r), where the summation is extended over all points in (0; x) at which some g ( ¢ ; n) is not c ..."
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Let g(x; n), with x 2 R+, be a step function for each n. Assuming certain technical hypotheses, we give a constant ® and function f such thatP1 n=1 g(x; n) can be written in the form ®+ P 0<r<x f(r), where the summation is extended over all points in (0; x) at which some g ( ¢ ; n) is not continuous. A typical example is P1 n=1 z bn=xc = 1