Ramsey goodness of trees in random graphs

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985807%3A_____%2F23%3A00561499" target="_blank" >RIV/67985807:_____/23:00561499 - isvavai.cz</a>
Result on the web
<a href="https://dx.doi.org/10.1002/rsa.21124" target="_blank" >https://dx.doi.org/10.1002/rsa.21124</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1002/rsa.21124" target="_blank" >10.1002/rsa.21124</a>

Result language
angličtina
Original language name
Ramsey goodness of trees in random graphs
Original language description
For graphs G, H and a family of graphs., we write G ->(H,P) to denote that every blue-red coloring of the edges of G contains either a blue copy of H, or a red copy of each F is an element of P For integers n and D, let tau(n, D) denote the family of all trees with n edges and maximum degree atmostD. We prove that for each r, D >= 2, there exist constants C, C' > 0 such that if p >= Cn(-2/(r+2)) and N >= rn + C'/p, then G(N, p)->(Kr+ 1, tau(n, D)) with high probability. This is a random version of a well-known result of Chvatal from 1977. The proof combines a stability argument with the embedding of trees in expander graphs. Furthermore, the proof of the stability result is based on a sparse random analogue of the Erd.os-Sos conjecture for trees with linear size and bounded maximum degree, which may be of independent interest.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
<a href="/en/project/GJ20-27757Y" target="_blank" >GJ20-27757Y: Random discrete structures</a><br>
Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2023
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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