Counting extensions revisited

0523428 - ÚI 2020 US eng V - Research Report
Šileikis, Matas - Warnke, L.
Counting extensions revisited.
Cornell University, 2019. 21 s. arXiv.org e-Print archive, arXiv:1911.03012 [math.CO].
R&D Projects: GA ČR(CZ) GA19-08740S
Institutional support: RVO:67985807
https://arxiv.org/abs/1911.03012

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. In 1989, Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices. For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.
Permanent Link: http://hdl.handle.net/11104/0307783