Letzter, S;
Sahasrabudhe, J;
(2020)
On existentially complete triangle-free graphs.
Israel Journal of Mathematics
, 236
(2)
pp. 591-601.
10.1007/s11856-020-1982-3.
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Abstract
For a positive integer k, we say that a graph is k-existentially complete if for every 0 ⩽ a ⩽ k, and every tuple of distinct vertices x1, …, xa, y1, …, yk−a, there exists a vertex z that is joined to all of the vertices x1, …, xa and to none of the vertices y1, …, yk−a. While it is easy to show that the binomial random graph Gn,1/2 satisfies this property (with high probability) for k = (1 − o(1)) log2n, little is known about the “triangle-free” version of this problem: does there exist a finite triangle-free graph G with a similar “extension property”? This question was first raised by Cherlin in 1993 and remains open even in the case k = 4. We show that there are no k-existentially complete triangle-free graphs on n vertices with k>8lognloglogn, for n sufficiently large.
| Type: | Article |
|---|---|
| Title: | On existentially complete triangle-free graphs |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s11856-020-1982-3 |
| Publisher version: | https://doi.org/10.1007/s11856-020-1982-3 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10107227 |
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