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

TL;DR

The paper proves a linear upper bound R(G, Γ) ≤ C·v(G) for zero-sum Ramsey numbers of n-vertex graphs G with bounded maximum degree Δ against any finite abelian group Γ whose order divides e(G). The authors develop a comprehensive framework built on blueprints (induced neighborhoods), gadgets (structured subgraphs), and realizations (maps embedding blueprint structures into gadgets), combined with an algorithm that operates through well-behaved colourings and strategic coset quotients to realize a zero-sum copy of G. The key innovations are the blueprint/gadget machinery and a three-phase algorithm that uses Kneser-type sumset arguments to force a zero-sum embedding, along with careful vertex and gadget-counting to maintain feasibility. These methods yield a robust approach that not only establishes the linear bound but also provides a blueprint for potential extensions to bounded degeneracy graphs and, conjecturally, to non-abelian groups, albeit with substantial additional challenges.

Abstract

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

Figures (8)

• Figure 1: Example of a $(2,2)$ gadget.
• Figure 2: An example of a $(9,7,2)$ blueprint pair, an induced subgraph of $G$. The dashed lines represent edges that are present in $G$, but do not affect the behavior of realizations of gadgets (see \ref{['sec.gadgets']}).
• Figure 3: Example of a $(9,7,2,3)$ gadget (the sets $P_v$ are not shown). In comparison to Figure \ref{['simple_gadget']}, instead of paths from $v$ to $u$ we have $6$ edges from $D_1$ to each $w_i$, $4$ edges from $D_2$ to each $w_j'$, 2 edges from $M$ to each $w_i$ and $w_j'$, and every edge $w_iw_j'$. This gadget corresponds to the blueprint in \ref{['blueprint_example']}.
• Figure 4: Example of a $(9,7,2,3)$--gadget (the sets $P_v$ are not shown) with $|X_2|=1$ and its reformulation as a $(9,3)$--gadget.
• Figure 5: An example of a $\kappa$--well-behaved vertex set, where the labels of vertices represent their colour under $\mathcal{C}$ and labels of edges represent $s+\mathcal{C} (x)+\mathcal{C} (y)$ for $x,y\in \{u,v,w\}$.

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: Chvátal, Ródl, Szemerédi and Trotter ChRSzT
• Theorem 3.1: Fundamental theorem of finite abelian groupsdummit2004
• Theorem 3.2: Kneser's Theorem Kneser_1953
• Lemma 3.3
• proof
• Definition 4.1
• Proposition 4.2
• Definition 5.1