arXiv subjects

Małgorzata Bednarska-Bzdȩga

Publications and source records attributed to Małgorzata Bednarska-Bzdȩga.

4 recordsLinked to original sources

arXiv2023

Off-diagonal online size Ramsey numbers for paths

Consider the following Ramsey game played on the edge set of $K_{\mathbb N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder's goal is to force Painter to create a red copy of a path $P_k$ on $k$ vertices or a blue copy of $P_n$ as soon as possible. The online (size) Ramsey number $\tilde{r}(P_k,P_n)$ is the number of rounds in the game provided Builder and Painter play optimally. We prove that $\tilde{r}(P_k,P_n)\le (5/3+o(1))n$ provided $k=o(n)$ and $n\to \infty$. We also show that $\tilde{r}(P_4,P_n)\le \lceil 7n/5\rceil -1$ for $n\ge 10$, which improves the upper bound obtained by J.~Cyman, T.~Dzido, J.~Lapinskas, and A.~Lo and implies their conjecture that $\tilde{r}(P_4,P_n)=\lceil 7n/5\rceil -1$.

math.CO↗

arXiv2023

Online Ramsey numbers: Long versus short cycles

Online Ramsey game is played between Builder and Painter on an infinite board $K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create either a red copy of $H_1$ or a blue copy of $H_2$ in $K_{\mathbb N}$ as soon as possible, and Painter tries to prevent it. The online Ramsey number $\tilde{r}(H_1,H_2)$ is the minimum number of rounds such that the Builder wins. We study $\tilde{r}(C_k,C_n)$ where $k$ is fixed and $n$ is large. We show that $\tilde{r}(C_k,C_n)=2n+\mathcal O(k)$ for an absolute constant $c$ if $k$ is even, while $\tilde{r}(C_k,C_n)\le 3n+o(n)$ if $k$ is odd.

math.CO↗

arXiv2019

On the separation conjecture in Avoider-Enforcer games

Given a fixed graph $H$ with at least two edges and positive integers $n$ and $b$, the strict $(1 \colon b)$ Avoider-Enforcer $H$-game, played on the edge set of $K_n$, has the following rules: In each turn Avoider picks exactly one edge, and then Enforcer picks exactly $b$ edges. Avoider wins if and only if the subgraph containing her/his edges is $H$-free after all edges of $K_n$ are taken. The lower threshold of a graph $H$ with respect to $n$ is the largest $b_0$ for which Enforcer has a winning strategy for the $(1\colon b)$ $H$-game played on $K_n$ for any $b \leq b_0$, and the upper threshold is the largest $b$ for which Enforcer wins the $(1 \colon b)$ game. The separation conjecture of Hefetz, Krivelevich, Stojaković and Szabó states that for any connected $H$, the lower threshold and the upper threshold of the Avoider-Enforcer $H$-game played on $K_n$ are not of the same order in $n$. Until now, the conjecture has been verified only for stars, by Grzesik, Mikalački, Nagy, Naor, Patkos and Skerman. We show that the conjecture holds for every connected graph $H$ with at most one cycle (and at least two edges), with a polynomial separation between the lower and upper thresholds. We also prove an upper bound for the lower threshold of any graph $H$ with at least two edges, and show that this bound is tight for all graphs in which each connected component contains at most one cycle. Along the way, we establish number-theoretic tools that might be useful for other problems of this type.

math.CO↗

arXiv2015

Picker-Chooser fixed graph games

Given a fixed graph $H$ and a positive integer $n$, a Picker-Chooser $H$-game is a biased game played on the edge set of $K_n$ in which Picker is trying to force many copies of $H$ and Chooser is trying to prevent him from doing so. In this paper we conjecture that the value of the game is roughly the same as the expected number of copies of $H$ in the random graph $G(n,p)$ and prove our conjecture for special cases of $H$ such as complete graphs and trees.

math.CO↗