Year: 2017
Type: preprint
Abstract: We prove that for each $D\ge 2$ there exists $c>0$ such that whenever $b\le c\big(\tfrac{n}{\log n}\big)^{1/D}$, in the $(1:b)$ Maker-Breaker game played on $E(K_n)$, Maker has a strategy to guarantee claiming a graph $G$ containing copies of all graphs $H$ with $v(H)\le n$ and $Δ(H)\le D$. We show further that the graph $G$ guaranteed by this strategy also contains copies of any graph $H$ with bounded maximum degree and degeneracy at most $\tfrac{D-1}{2}$. This lower bound on the threshold bias is sharp up to the $\log$-factor when $H$ consists of $\tfrac{n}{3}$ vertex-disjoint triangles or $\tfrac{n}{4}$ vertex-disjoint $K_4$-copies.
Source: arXiv (Cornell University)
Authors: Allen, Peter, Böttcher, Julia, Kohayakawa, Yoshiharu, Naves, Humberto, Person, Yury
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G171752294, G1823433807, G2408798171, G2777201912, G3034931302, G3347278549, G3502892431, G3746528247, G4174510764, G4608934693, G4643350488, Ramsey Probleme in zufälligen und zufällig perturbierten diskreten Strukturen, und deren scharfe Schwellenwerte, G5341864644, G5577358074, G6454881139, G6739720741, G6812024786, G8539280978
OpenAlex ID: W2771027095