{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/path-monochromatic-bounded-depth-rooted-trees","title":"Path-monochromatic bounded depth rooted trees in (random) tournaments","arxiv_id":"2404.03752","date":"2024-04-04","proceeding":null,"authors":["Raphael Yuster"],"abstract":"An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\\ell$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge coloring of $T$ has a path-monochromatic subtree with at least $f_T(k)$ vertices and let $f_T(k,\\ell)$ be the restriction to subtrees of depth at most $\\ell$. It was proved by Landau that $f_T(1,2)=n$ and proved by Sands et al. that $f_T(2)=n$ where $|V(T)|=n$. Here we consider $f_T(k)$ and $f_T(k,\\ell)$ in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-H\\\"aggkvist Conjecture. We also study the typical value of $f_T(k)$ and $f_T(k,\\ell)$, i.e., when $T$ is a random tournament.","url_abs":"https://arxiv.org/abs/2404.03752v1","url_pdf":"https://arxiv.org/pdf/2404.03752v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"path-monochromatic-bounded-depth-rooted-trees","repo_url":"https://github.com/raphaelyuster/path-monochromatic","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}