Odd-Ramsey Numbers in Extremal Combinatorics

• Odd-Ramsey number is a combinatorial invariant defined as the minimum number of colors needed so every copy of a subgraph contains a color class with an odd number of edges.
• It generalizes classical Ramsey theory by replacing the monochromatic condition with a parity constraint, leading to novel bounds and asymptotic estimates.
• The theory leverages algebraic, coding theoretic, and probabilistic methods, with applications in combinatorial coding theory and parity-obstructed network design.

The odd-Ramsey number is a recently formalized invariant at the intersection of Ramsey theory and parity-based extremal combinatorics. For given host and pattern graphs (or hypergraphs) $G$ and $H$, the odd-Ramsey number $r_{\mathrm{odd}}(G,H)$ denotes the minimal number of colors needed for an edge-coloring of $G$ such that every copy of $H$ contains a color class whose intersection with $E(H)$ has odd cardinality. This parameter generalizes classical Ramsey numbers by enforcing a parity obstruction on subgraphs, replacing the requirement of monochromaticity with that of odd intersection. The theory of odd-Ramsey numbers is under rapid development, motivated by applications in combinatorial coding theory, parity-obstructed network design, and generalized extremal problems, and connects deeply with symmetry-breaking and error-correcting code constructions.

1. Formal Definition and Variants

Let $G$ be a finite (hyper)graph, $H\subseteq G$ a fixed (hyper)subgraph. An $r$-edge-coloring of $G$ is said to be $H$-odd if every $C\cong H$ in $G$ has $|E(C)\cap E_i|$ odd for some color class $E_i$. The odd-Ramsey number is

$r_{\mathrm{odd}}(G,H) := \min \left\{ r : \text{there exists an %%%%15%%%%-coloring of %%%%16%%%% that is %%%%17%%%%-odd} \right\}.$

For families $\mathcal{H}$ of subgraphs (e.g., all spanning $K_{t,n-t} \subset K_n$), write $r_{\mathrm{odd}}(n,\mathcal{H})$ for the minimum $r$ such that every $H\in\mathcal{H}$ is odd-colored in some class in every coloring of $K_n$ (Boyadzhiyska et al., 8 Oct 2024).

If $H$ has an odd number of edges, then $r_{\mathrm{odd}}(G,H) = 1$, as trivial parity guarantees an odd color count in any coloring. All meaningful cases focus on $H$ with even $|E(H)|$.

This concept extends to hypergraphs; for uniform $k$-graphs $G$ and $H$ the odd-Ramsey number counts the minimum $r$ for edge-colorings of $G$ so that in every copy of $H$, some color class occurs an odd number of times (Crawford et al., 25 Jul 2025).

2. Odd-Ramsey Numbers: Main Results in Graphs

Hamilton Cycles: For $C_n$ denoting the $n$-cycle with $n$ even, the odd-Ramsey number of the Hamilton cycle is tightly bracketed as

$$\left(\tfrac{\sqrt{2}}{2} + o(1)\right)\sqrt{n} \leq r_{\text{odd}}(n,C_n) \leq \tfrac{3\sqrt{2}}{2}\sqrt{n},$$

with constants arising from explicit finite-field constructions (upper bound) and combinatorial parity-switch arguments (lower bound) (Boyadzhiyska et al., 13 Nov 2025).

Spanning Complete Bipartite Graphs: For the family $\mathcal{F}$ of all spanning $K_{t,n-t}$ in $K_n$,

$r_{\rm odd}(n,\mathcal{F}) = \begin{cases} n-1 & \text{if %%%%40%%%% even,} \ n & \text{if %%%%41%%%% odd.} \end{cases}$

This resolves the value exactly and exploits Chevalley–Warning-type counting arguments (Boyadzhiyska et al., 8 Oct 2024).

Fixed Bipartite Subgraphs: For fixed $K_{s,t}$ with $st$ even,

$$r_{\rm odd}(n,K_{s,t}) \ge (1+o(1)) n^{1/\lfloor s/2 \rfloor}, \quad r_{\rm odd}(n,K_{2,t}) = \Theta(n).$$

The lower bound follows from double-counting arguments adapting classical Kővári–Sós–Turán theory, while the upper bound utilizes generalized Ramsey numbers (Boyadzhiyska et al., 8 Oct 2024).

3. Methods: Algebraic, Coding Theoretic, and Probabilistic Tools

Finite-Field and Algebraic Constructions: For Hamilton cycles, finite-field labelings and associated color palettes yield explicit colorings avoiding "even-colored" cycles, delivering constructive upper bounds. In the setting $n=m2^t$, label $V= \mathbb{F}_2^t \times [m]$ and color edges so sums of coordinates force odd color classes in any $C_n$ (Boyadzhiyska et al., 13 Nov 2025).

Parity-Switch (Switch-Merging) Framework: The lower bound involves iterative merging of color classes via specially structured 4-cycles ("switches") enabling reduction to a single-color scenario and controlling even-parity Hamilton cycles (Boyadzhiyska et al., 13 Nov 2025).

Coding-Theoretic Duality: For bipartition families, the problem is equivalent to maximizing the dimension $\ell(n, W_T)$ of a binary linear code of length $n$ that avoids codewords of weight in a forbidden set $W_T$, leading to

$r_{\rm odd}(n, \mathcal{F}_T) = n - \ell(n, W_T)$

for appropriate weight sets $W_T$ derived from bipartition sizes (Boyadzhiyska et al., 8 Oct 2024).

Probabilistic and Hypergraph Matching Arguments: For multipartite hosts $K_{n,n}$ and $K_{2,t}$ subgraphs, as well as $k$-uniform hypergraphs, the upper bounds exploit randomized or conflict-free hypergraph-matching theorems (e.g., Tripartite Matching Theorem of Joos–Mubayi–Smith) to demonstrate the existence of suitable colorings with the correct asymptotic behavior (Crawford et al., 25 Jul 2025).

4. Asymptotic, Exact, and Coding-Theoretic Results

For large parameter regimes, asymptotic and, in several cases, exact values of the odd-Ramsey number have been determined:

Pattern $H$	Host $G$	$r_{\mathrm{odd}}(G,H)$	Reference
Hamilton cycle $C_n$	$K_n$	$\Theta(\sqrt{n})$	(Boyadzhiyska et al., 13 Nov 2025)
All spanning $K_{t,n-t}$	$K_n$	$n$ (odd $n$), $n-1$ (even $n$)	(Boyadzhiyska et al., 8 Oct 2024)
$K_{2,t}$	$K_{n,n}$	$n/t + o(n)$	(Crawford et al., 25 Jul 2025)
$_{1,...,1,2,2}$, $k$-uniform	$K_{n,\dots,n}$	$n/2 + o(n)$	(Crawford et al., 25 Jul 2025)

In the bipartite setting, the link to binary codes provides tight bounds: for subfamilies $\mathcal{F}_T$ specified by $T$ (set of bipart sizes with $t(n-t)$ even),

$r_{\rm odd}(n,\mathcal{F}_T) = n - \ell(n,W_T),$

enabling transfer of classical and new coding bounds directly into Ramsey-type extremal results (Boyadzhiyska et al., 8 Oct 2024).

5. Relation to Classical Ramsey Theory and Codes

The odd-Ramsey number diverges fundamentally from standard Ramsey numbers. Classical diagonal Ramsey for non-bipartite graphs queries the minimal $N$ forcing a monochromatic $H$. Here, the requirement is to force one color class to appear oddly within each copy of $H$—monochromaticity is sufficient, but not necessary.

This relaxation brings the odd-Ramsey theory close to anti-Ramsey or coloring-type extremal problems, exploiting symmetry and anti-parity. Moreover, the equivalence for bipartite patterns and large $n$,

$r_{\rm odd}(n,\mathcal{F}_T) = n - \ell(n,W_T)$

connects the extremal coloring problem to the maximal size of codes with forbidden weight spectrum—a direct combinatorial duality (Boyadzhiyska et al., 8 Oct 2024).

The finite-field construction for Hamilton cycles is reminiscent of code constructions in the design of error-detecting and -correcting systems, where odd intersections correspond to "detectability" of certain error patterns (Boyadzhiyska et al., 13 Nov 2025).

6. Extensions: Hypergraphs and Multipartite Hosts

The theory generalizes naturally to hypergraphs and multipartite graph hosts. The results in $k$-uniform, complete $k$-partite hypergraphs establish that

$$r_{\mathrm{odd}}\left(\mathcal{K}^{(k)}_{n,\dots,n}, \mathcal{K}_{1,\dots,1,2,2}\right) = \frac{n}{2} + o(n),$$

the first such asymptotic for hypergraph-host odd-Ramsey numbers (Crawford et al., 25 Jul 2025).

For fixed complete bipartite graphs $K_{s,t}$,

$$r_{\rm odd}(n,K_{s,t}) = \Omega\left(n^{1/\lfloor s/2\rfloor}\right),$$

demonstrating diverse asymptotic behaviors depending on the pattern size; for $s=2$, the regime is linear, while for larger $s$, growth may be sublinear but super-polylogarithmic (Boyadzhiyska et al., 8 Oct 2024).

7. Open Problems and Research Directions

Several challenging questions remain:

• Determining exact constants in the leading terms for $r_{\mathrm{odd}}(n,H)$, particularly for cycles and small bipartite graphs.
• Identifying explicit (deterministic or algebraic) constructions matching the probabilistic upper bounds for general host–pattern pairs, especially in higher uniformity or multipartite hypergraphs (Crawford et al., 25 Jul 2025).
• Establishing whether generalizations to other modulus constraints (e.g., $|E(H) \cap E_i| \equiv r \pmod m$) yield qualitatively new phenomena or connections to higher-order coding theory.
• Characterizing the precise range of $n$ for which the exact results for Hamilton cycles and spanning bipartite subgraphs hold; for small $n$ relative to $|H|$, behavior may deviate from the asymptotic regime (Boyadzhiyska et al., 13 Nov 2025, Boyadzhiyska et al., 8 Oct 2024).
• Uncovering further connections between parity-type Ramsey numbers and both linear and non-linear coding invariants.

The odd-Ramsey number thus represents a rich intersection of extremal combinatorics, algebraic constructions, probabilistic methods, and information theory.