Resolving the Kohayakawa--Kreuter Conjecture for Families
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910177087717376 |
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| author | Yancey, Matthew |
| author_facet | Yancey, Matthew |
| contents | A graph $G$ is $(a,b)$-sparse if every nonempty subgraph $H$ satisfies $e(H) \leq a v(H) - b$. We are interested in the conditions under which an $(a,b)$-sparse graph can be partitioned $E(G) = E(G_1) \cup E(G_2)$ such that for $i \in \{1,2\}$ we have that $G_i$ is $(a_i, b_i)$-sparse. Kuperwasser, Samotij, and Wigderson conjectured that a $(m,0)$-sparse graph can be partitioned into a $(1,1)$-sparse graph and a $(m,2m-1)$-sparse graph. We prove the conjecture in full.
The Kohayakawa--Kreuter Conjecture for Families claims that $n^{-1/m_2}$ is the threshold function for the random graph being Ramsey a.a.s. for graph families $\mathcal{H}_1, \ldots \mathcal{H}_r$. Kuperwasser, Samotij, and Wigderson motivated their conjecture by proving that it is sufficient to establish the Kohayakawa--Kreuter Conjecture for Families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_03086 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resolving the Kohayakawa--Kreuter Conjecture for Families Yancey, Matthew Combinatorics A graph $G$ is $(a,b)$-sparse if every nonempty subgraph $H$ satisfies $e(H) \leq a v(H) - b$. We are interested in the conditions under which an $(a,b)$-sparse graph can be partitioned $E(G) = E(G_1) \cup E(G_2)$ such that for $i \in \{1,2\}$ we have that $G_i$ is $(a_i, b_i)$-sparse. Kuperwasser, Samotij, and Wigderson conjectured that a $(m,0)$-sparse graph can be partitioned into a $(1,1)$-sparse graph and a $(m,2m-1)$-sparse graph. We prove the conjecture in full. The Kohayakawa--Kreuter Conjecture for Families claims that $n^{-1/m_2}$ is the threshold function for the random graph being Ramsey a.a.s. for graph families $\mathcal{H}_1, \ldots \mathcal{H}_r$. Kuperwasser, Samotij, and Wigderson motivated their conjecture by proving that it is sufficient to establish the Kohayakawa--Kreuter Conjecture for Families. |
| title | Resolving the Kohayakawa--Kreuter Conjecture for Families |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.03086 |