arXiv subjects

Jasmin Katz

Publications and source records attributed to Jasmin Katz.

3 recordsLinked to original sources

arXiv2026

A degree version of the Burr-Erd\H{o}s conjecture on trees

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erd\H{o}s conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

math.CO↗

arXiv2025

A linear upper bound for zero-sum Ramsey numbers of bounded degree graphs

Let $G$ be a graph and $\Gamma$ a finite abelian group. The zero-sum Ramsey number of $G$ over $\Gamma$, denoted by $R(G, \Gamma)$, is the smallest positive integer $t$ (if it exists) such that any edge-colouring $c:E(K_t)\to\Gamma$ contains a copy of $G$ with $\sum_{e\in E(G)}c(e)=0_\Gamma$. We prove a linear upper bound $R(G, \Gamma)\leq Cn$ that holds for every $n$-vertex graph $G$ with bounded maximum degree and every finite abelian group $\Gamma$ with $|\Gamma|$ dividing $e(G)$.

math.CO↗

arXiv2025

Bounds for Hypergraph Universality

A graph $\Gamma$ is said to be universal for a class of graphs $\mathcal{H}$ if $\Gamma$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $\Gamma$ to be universal for the class of $D$-degenerate graphs on $n$ vertices has been shown to be $O(n^{2-1/D}(\log n)^{2/D}(\log\log n)^{5})$. We generalise this result to $r$-uniform hypergraphs, showing the following. Given $D, r \ge 2$ and $n$ sufficiently large, there exists a constant $C = C(D, r)$ such that there exists a graph with at most \[Cn^{r-1/D}(\log n)^{2/D}(\log\log n)^{2r+1}\] edges which is universal for the class of $D$-degenerate $r$-uniform hypergraphs on $n$ vertices. This is tight up to the polylogarithmic term.

math.CO↗