A central challenge in network design is ensuring resilience: how can we guarantee multiple, independent, communication pathways between nodes, even when some connections fail in a network? In 1989, Zehavi and Itai formulated a graph-theoretic conjecture that captures the essence of this problem. They proposed that any \(k\)-vertex-connected graph contains \(k\) independent spanning trees rooted at any given root \(r\), which means that for every vertex \(v\) in the graph, the unique \(r-v\) paths within these \(k\) spanning trees are entirely disjoint, apart from their endpoints \(r\) and \(v\). Despite decades of effort, this conjecture has only been proven for \(k \le 4\) and for specific graph families using their underlying topological structure, leaving the general case as an open problem in graph theory with substantial consequences in the field of distributed algorithms.

We present new results on Maker-Breaker games arising from the Erdős-Szekeres problem in planar geometry. This classical problem asks how large a set in general position has to be to ensure the existence of n points that are the vertices of a convex n-gon. Moreover, Erdős further extended this problem by asking what happens if we also require that this n-gon has an empty interior. In a 2-player Maker-Breaker setting, this problem inspires two main games. In both games, Maker tries to obtain an empty convex k-gon, while Breaker tries to prevent her from doing so. The games differ only in which points can comprise the winning k-gons: in the monochromatic version the points of both players can make up a k-gon, while in the bichromatic version only Maker’s points contribute to such a polygon. Both settings are studied in this paper. We show that in the monochromatic game, Maker always wins. Even in a biased game where Breaker is allowed to place s points per round, for any constant $$s \ge 1$$ , Maker has a winning strategy. In the bichromatic setting, Maker still wins whenever Breaker is allowed to place s points per round for any constant $$s<2$$ . This settles an open problem posed in 2019. Furthermore, we show that there are games that are not a lost cause for Breaker. Whenever $$k\ge 8$$ and Breaker is allowed to play 12 or more points per round, she has a winning strategy. We also consider the one-round bichromatic game (a.k.a. the offline version). In this setting, we show that Breaker wins if she can place twice as many points as Maker but if the bias is less than 2, then Maker wins for large enough set of points.

Confirming a conjecture of Gyárfás, we prove that, for all natural numbers $k$ and $r$, the vertices of every $r$-edge-colored complete $k$-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers $p$ and $r$, the vertices of every $r$-edge-colored complete graph can be partitioned into a bounded number of $p$th powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklóssy [ Discrete Math., 340 (2017), pp. 2053--2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number.

Given a pair of graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the smallest $N$ such that every red-blue coloring of the edges of the complete graph $K_N$ contains a red copy of $G$ or a blue copy of $H$. If a graph $G$ is connected, it is well known and easy to show that $R(G,H) \geq (|G|-1)(\chi(H)-1)+\sigma(H)$, where $\chi(H)$ is the chromatic number of $H$ and $\sigma(H)$ is the size of the smallest color class in a $\chi(H)$-coloring of $H$. A graph $G$ is called $H$-good if $R(G,H)= (|G|-1)(\chi(H)-1)+\sigma(H)$. The notion of Ramsey goodness was introduced by Burr and Erdös in 1983 and has been extensively studied since then. In this paper we show that if $n\geq 10^{60}|H|$ and $\sigma(H)\geq \chi(H)^{22}$, then the $n$-vertex cycle $C_n$ is $H$-good. For graphs $H$ with high $\chi(H)$ and $\sigma(H)$, this proves in a strong form a conjecture of Allen, Brightwell, and Skokan.