An exact robust Ramsey theorem for matchings

Abstract: Keevash and Michaeli recently proved that, under the robustness assumption that (G) is an (s)-connector (i.e. (\overline G) is (K_{s,s})-free), (G) has essentially the same multicolour Ramsey matching properties as complete graphs, with an additive error (O(qs)), where (q) is the number of colours. They asked whether the dependence on (q) can be removed. We answer this question in a stronger exact form. For ({\bf t}=(t_1,\ldots,t_q)\in\mathbb N_+^q), let (R_s({\bf t})) be the smallest integer (N) such that every (N)-vertex (s)-connector (G) satisfies ( G\to (t_1K_2,\ldots,t_qK_2). ) We determine the exact value [ R_s({\bf t})=\sum_{j\in[q]}(t_j-1)+ \max\left{2s,\ s+\max_{j\in[q]}t_j\right}. ] While Keevash and Michaeli's proof uses a compression algorithm based on the Gallai--Edmonds decomposition to reduce the colouring to a structured form, our proof is a direct minimal-counterexample argument together with a new counting method for monochromatic matchings which can be applied to (s)-connectors.

• The paper achieves an exact determination of the robust Ramsey threshold for matchings in s-Connectors, eliminating color dependency.
• It utilizes a minimal-counterexample strategy combined with new counting methods to derive explicit bounds expressed via s and matching sizes.
• The result generalizes the Cockayne-Lorimer theorem and sharpens our understanding of structural obstructions in sparse graph Ramsey theory.

Exact Thresholds for Ramsey Matchings in $s$-Connectors

Overview

This paper establishes the exact threshold for multicolor Ramsey matchings in $s$-connectors, a combinatorial class characterized by the absence of $(s,s)$ bipartite holes. The authors solve an outstanding question posed by Keevash and Michaeli regarding the elimination of the color dependency in robust Ramsey matching bounds, demonstrating that the resulting threshold is not merely independent of the number of colors $q$, but can be expressed in explicit form. This result precisely quantifies the robust loss relative to complete graphs and clarifies the structural obstructions in the problem.

Context and Motivation

Ramsey-type extremal results analyze the conditions under which every edge-coloring of a host graph yields a monochromatic copy of a prescribed subgraph. Traditionally, these thresholds are evaluated in complete graphs, notably described by the Cockayne-Lorimer theorem for matchings. Keevash and Michaeli extended the theory to $s$-connectors, quantifying the robust Ramsey threshold with an additive term $O(qs)$ and conjectured the possibility of eliminating $q$ altogether. The $s$-connector property replaces the usual independence number obstruction with a bipartite hole criterion, producing a more natural framework for matching Ramsey properties.

Main Results

The primary contribution is the determination of the exact robust Ramsey threshold for matchings in $s$-connectors. For prescribed non-negative integers $\mathbf{t} = (t_1, \dots, t_q)$, the minimal $s$0 such that every $s$1-vertex $s$2-connector $s$3 satisfies $s$4 is given by:

$s$5

where $s$6 and $s$7.

This eliminates all dependence on $s$8 in the bound. This threshold is shown to be tight, with explicit lower-bound constructions corresponding to both terms in the maximum. The result recovers the classical Cockayne-Lorimer theorem when $s$9, and quantifies the exact loss encountered when incomplete host graphs ($(s,s)$0-connectors) are used.

Corollary 1.4 further links the Ramsey threshold to the bipartite independence number $(s,s)$1, providing a tight lower bound:

$(s,s)$2

for graphs $(s,s)$3 which do not satisfy the multicolor matching Ramsey property.

Methodology

The proof diverges from prior works reliant on compression algorithms and structural decompositions (notably the Gallai-Edmonds decomposition) and is based on a minimal-counterexample approach bolstered by a new counting method for monochromatic matchings in $(s,s)$4-connectors. Core techniques include:

• Structural Analysis of Minimal Counterexamples: Monochromatic components are shown to be factor-critical, while the component hypergraph is acyclic. This restricts potential obstructions to a tree-like arrangement around a largest monochromatic piece.
• Component Hypergraph and Incidence Graphs: Overlap between monochromatic components is encoded using hypergraphs, which are then analyzed via their incidence bipartite graphs.
• Branch Analysis and Matching Contributions: The tree structure enables precise quantification of the matching contribution from branches outside the largest monochromatic component, leading directly to the sharp upper bound.
• Separator Lemmas and Centroid Arguments: Separator results for acyclic hypergraphs with bounded hyperedge size play a central role in limiting the size of branches and their contribution to the obstruction.

Numerical Strengths and Claims

• The threshold $(s,s)$5 is both necessary and sufficient, with explicit constructions showing that neither the $(s,s)$6 nor the $(s,s)$7 terms in the maximum can be omitted.
• For all $(s,s)$8, the exact threshold holds, confirming the conjecture and considerably improving previous bounds (eliminating the $(s,s)$9 error term).

Theoretical and Practical Implications

The exact threshold for Ramsey matchings in $q$0-connectors sharpens the understanding of robust Ramsey-type properties in sparse graphs. The result demonstrates that, for matchings, the robust loss is governed by the maximum of two unavoidable obstructions:

1. The bipartite hole (quantified as $q$1).
2. The largest matching size plus the connector parameter ($q$2).

This advances combinatorial transference, from complete graphs to random graphs and now to deterministic sparse structures. The explicit formula enables direct application in algorithmic contexts, especially in extremal graph theory or randomized settings where $q$3-connector properties are prevalent.

Future Directions

The authors outline several avenues for further investigation:

• Stability and Classification of Extremal Obstructions: Characterizing all tight examples at the threshold remains open.
• Host Graphs with Weaker Properties: Extending results to graphs sparser than $q$4-connectors (e.g., those only meeting certain degree or expansion conditions).
• Hypergraph Analogues: While the method is tightly coupled to factor-critical components, extending the theory to hypergraphs poses significant challenges, potentially requiring new structural frameworks.

Conclusion

This work resolves a pivotal question in robust Ramsey theory for matchings, providing an exact threshold in $q$5-connectors. The combinatorial and structural approach employed not only confirms and improves previous conjectures but also offers a precise and operational formula for assessing Ramsey thresholds in sparse host graphs. The results clarify the role of bipartite holes and matching size, suggesting broader applications and further generalizations within extremal and probabilistic combinatorics.