PreprintCC BY 4.0arXiv
The Erd\H os-Sós conjecture in dense graphs
Abstract
The Erd\H os--Sós conjecture states that every n-vertex graph with more than (k−2)n/2 edges contains every k-vertex tree. We prove that for every γ there is an n0 such that for all n≥ n0 and k ≥ γn the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and Graham on the multicolor Ramsey numbers of trees.
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BibTeX
@misc{reed2026erdh,
title = {{The Erd\H os-Sós conjecture in dense graphs}},
author = {Bruce Reed and Maya Stein},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05417},
eprint = {2609.05417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05417},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Erd\H os-Sós conjecture in dense graphs
AU - Bruce Reed
AU - Maya Stein
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.05417
UR - https://arxiv.org/abs/2609.05417
AB - The Erd\H os--Sós conjecture states that every n-vertex graph with more than (k−2)n/2 edges contains every k-vertex tree. We prove that for every γ there is an n_0 such that for all n≥ n_0 and k ≥ γn the conjecture holds. As a corollary of our result, we obtain a solution of a 51-year-old problem of Erd\H os and Graham on the multicolor Ramsey numbers of trees.
ER -