Thresholds for constrained Ramsey and anti-Ramsey problems

TL;DR

This work determines 0-statements for the constrained Ramsey property in random graphs when the first graph is a star $K_{1,k}$ with $k\ge 3$ and the second graph $H_2$ is not a forest, complementing existing 1-statements to nearly complete the picture except the $H_1=K_{1,2}$ case. It develops a reduction framework to colouring statements that control anti-Ramsey-type thresholds, and introduces triangle-connected structures to handle the triangle case $H_2=K_3$, providing explicit constructive colourings that avoid rainbow copies of $H_2$ while preventing monochromatic $K_{1,k}$, thus establishing the conjectured thresholds up to coarse precision. For the anti-Ramsey problem, the paper reduces the 0-statement to a necessary colouring statement for strictly $2$-balanced graphs and proves it in several families, using $H$-blocks and $H$-closed decompositions alongside probabilistic tools like Janson’s inequality to derive lower bounds. The results significantly advance the understanding of thresholds for constrained Ramsey and anti-Ramsey properties in $G(n,p)$, clarifying when the natural densities $n^{-1/m_2(H)}$ govern the transition and highlighting remaining open cases and potential sharp-threshold phenomena.

Abstract

Let $H_1$ and $H_2$ be graphs. A graph $G$ has the constrained Ramsey property for $(H_1,H_2)$ if every edge-colouring of $G$ contains either a monochromatic copy of $H_1$ or a rainbow copy of $H_2$. Our main result gives a 0-statement for the constrained Ramsey property in $G(n,p)$ whenever $H_1 = K_{1,k}$ for some $k \ge 3$ and $H_2$ is not a forest. Along with previous work of Kohayakawa, Konstadinidis and Mota, this resolves the constrained Ramsey property for all non-trivial cases with the exception of $H_1 = K_{1,2}$, which is equivalent to the anti-Ramsey property for $H_2$. For a fixed graph $H$, we say that $G$ has the anti-Ramsey property for $H$ if any proper edge-colouring of $G$ contains a rainbow copy of $H$. We show that the 0-statement for the anti-Ramsey problem in $G(n,p)$ can be reduced to a (necessary) colouring statement, and use this to find the threshold for the anti-Ramsey property for some particular families of graphs.

Figures (3)

• Figure 1: The implementation of algorithm Rainbow-colour.
• Figure 2: The algorithm Rainbow-colour-constrained.
• Figure 4: The graph $G$ in the statement of Lemma \ref{['lem:tight_triangle']}

Theorems & Definitions (46)

• Theorem 1.2: anti-ram_1-statement
• Theorem 1.3
• Theorem 1.4
• Theorem 1.6
• Lemma 2.2
• proof
• Definition 2.3: $H$-equivalence
• Definition 2.4: $H$-closed property
• Definition 2.5: $H$-blocks
• Theorem 2.6: npss