Rainbow Turán Number: Concepts & Results

• Rainbow Turán number is a key parameter in extremal combinatorics that quantifies the maximum edges in properly edge-colored graphs while avoiding rainbow copies of a prescribed subgraph.
• The study extends classical Turán theory by incorporating edge-color restrictions, resulting in novel asymptotic bounds and creative graph constructions.
• Recent research employs augmentation and expander methods to derive sharp bounds and distinguish behaviors for various classes, including cycles, trees, and cliques.

A rainbow Turán number is a central parameter in extremal combinatorics, quantifying the maximal edge density in properly edge-colored graphs that avoid rainbow copies of a prescribed subgraph. Originating in work by Keevash, Mubayi, Sudakov, and Verstraëte (2007), the concept extends classical Turán-type extremal questions to the colored setting, introducing phenomena unique to the interplay between coloring and extremal graph structure. The study of rainbow Turán numbers has led to sharp results and novel methods for a wide class of host graphs and forbidden subgraphs, and continues to reveal intricate distinctions from their uncolored analogues.

1. Definitions and Fundamental Properties

Let $F$ be a fixed (uncolored) graph. For an integer $n$, the rainbow Turán number $\ex^*(n, F)$ is

$\ex^*(n, F) = \max\left\{\,|E(G)| : |V(G)| = n,\, G\, \text{properly edge-colored},\, G\, \text{has no rainbow copy of}\, F \right\},$

meaning $G$ is a graph on $n$ vertices, edges assigned colors so that adjacent edges have distinct colors, and G contains no subgraph isomorphic to $F$ whose edges all receive different colors (Jiang et al., 2021, Bednar et al., 2022, Janzer et al., 2022, Halfpap, 17 Dec 2025, Gerbner et al., 2019).

For comparison, the ordinary Turán number $\ex(n, F)$ is the maximal number of edges in an $n$-vertex graph containing no (not necessarily rainbow) copy of $F$, and trivially $\ex^*(n, F) \geq \ex(n, F)$.

A rainbow subgraph of $G$ is a subgraph whose edges all have distinct colors. The study naturally generalizes to the generalized rainbow Turán number, $\ex(n, H,\, \text{rainbow-}F)$: the maximal number of copies of $H$ in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$ (Balogh et al., 2020, Janzer, 2020, Gerbner et al., 2019).

2. Classical Results and Main Asymptotics

The initial study [Keevash–Mubayi–Sudakov–Verstraëte 2007] established fundamental dichotomies:

• Non-bipartite $F$: $\ex^*(n,F) = (1+o(1))\,\ex(n,F)$ as $n\to\infty$.
• Bipartite $F$: Behavior diverges; for example, for even cycles $C_{2k}$,

$\ex^*(n,C_{2k}) = \Theta(n^{1+1/k}),$

settling a conjecture in (Janzer, 2020, Das et al., 2012, Janzer et al., 2022, Jiang et al., 2021).

For paths $P_\ell$ of length $\ell$, the best known bounds are linear: $(k/2)\,n + O(1) \leq \ex^*(n, P_{k+1}) < (9k/7 + 2)n\quad \text{for all}\ k \ (\text{with improved constants for small }k),$ and $\ex^*(n, P_5) = \frac{5}{2} n + O(1)$ has been determined exactly (Halfpap, 2022, Ergemlidze et al., 2018, Johnston et al., 2019, Johnston et al., 2016).

For trees more generally, tight bounds depend on structure (stars, double stars, caterpillars, brooms). For example, if $F$ is a double star $DS_{r,s}$,

$(s+r-1)\frac{n}{2} + o(n) \leq \ex^*(n, DS_{r,s}) \leq (s+2r)\frac{n}{2},$

and for brooms $B_{k,2}$ and $B_{t,3}$, asymptotic formulas have been established with subtle dependencies on $k$ and divisibility (Halfpap, 17 Dec 2025, Byrne et al., 22 Feb 2025, Bednar et al., 2022).

For cycles, recent progress provides sharp bounds:

• For cycles $C_k$, up to logarithmic factors,

$n \log n \leq \ex^*(n, C_k) \leq c\, n (\log n)^2$

with precise results for even cycles, e.g.,

$\ex^*(n, C_{2k}) = \Theta(n^{1+1/k})$

(Janzer, 2020, Janzer et al., 2022, Wang, 2022, Balogh et al., 2020).

3. Advanced Techniques: Augmentation, Expander Methods, and Densities

The "augmentation" or "reduction" method (Bednar et al., 2022) constructs for a given $H$ an augmented graph $H'$ such that every proper edge-coloring of $H'$ forces a rainbow copy of $H$: $\ex^*(n, H) \leq \ex(n, H'),$ where $\ex(n, H')$ is the ordinary Turán number. If $H'$ is a tree, then the Erdős–Sós conjecture provides $\ex(n, H') \leq (e(H')-1)n/2$.

For forbidden rainbow clique subdivisions, the robust colored expander method (Sudakov–Tomon framework) can be adapted. For a fixed $t$, if $G$ is a properly edge-colored graph with

$e(G) \geq n \exp(c \sqrt{\log n}) = n^{1+o(1)},$

then $G$ contains a rainbow subdivision of $K_t$ with all replacement paths of logarithmic squared length, which is sharp up to the $o(1)$ term (Jiang et al., 2021). Key ingredients include minimal subgraph decompositions, expansion properties in color-restricted settings, and the strategic avoidance of forbidden colors and vertices during rainbow path construction.

The rainbow Turán density for graphs or trees in the graphon-system framework, introduced for trees in (Im et al., 2023), is defined as

$\pi^*_k(H) = \lim_{n \to \infty} \frac{\ex^*_k(n,H)}{\binom{n}{2}}$

and can be irrational or algebraic, contrasting with classical Turán densities which are rational of the form $1 - 1/t$. Stars uniquely maximize the density among all $k$-edge trees: $\pi_k^*(K_{1,k}) = \left(\frac{k-1}{k}\right)^2.$

4. Exact Bounds and Constructions for Small Graphs

For forests of stars, the extremal structures have been explicitly characterized. For star-forests without isolated edges, the extremal number is

$\ex^*(n, F) = \left\lfloor \frac{(e(F)-1)n}{2} \right\rfloor + O(1).$

For matchings $M_k$, exact formulas are available for large $n$ (Johnston et al., 2016).

For small paths and cycles, constructions such as vector-difference colorings of $K_{2^s}$ and Hamming graphs yield the exact lower bounds, e.g. for $P_k$ and certain small broom and caterpillar trees (Johnston et al., 2019, Halfpap, 17 Dec 2025, Byrne et al., 22 Feb 2025).

The table summarizes leading terms for rainbow Turán number of several classes:

Forbidden Subgraph	Asymptotic Bound/Formula	Reference
Path $P_k$	$\frac{k}{2}\,n + O(1)$ (lower),	(Johnston et al., 2019, Halfpap, 2022)
	$(9k/7+2)\,n$ (upper)	(Ergemlidze et al., 2018)
Double Star $DS_{r,s}$	$$(s+r-1)\frac{n}{2} \leq ex^* \leq (s+2r)\frac{n}{2}$$	(Bednar et al., 2022)
Broom $B_{k,2}$	see piecewise formula in (Halfpap, 17 Dec 2025)	(Halfpap, 17 Dec 2025)
Cycle $C_{2k}$	$\Theta(n^{1+1/k})$	(Janzer, 2020, Janzer et al., 2022)

5. Rainbow Turán Problems for Cycles, Subdivisions, and Generalized Variants

For cycles, the best known upper bound is

$\ex^*(n, C_k) < 8 n (\log n)^2,$

as shown by Janzer–Sudakov by combining spectral techniques, weighted homomorphism counts, and iterative minimization of color-collision contributions (Janzer et al., 2022). The classic construction of the $m$-cube with edge coloring by coordinate directions (no rainbow cycle) yields the lower bound $\Omega(n \log n)$.

For subdivisions of cliques, if $G$ is any properly edge-colored $n$-vertex graph with $n^{1+o(1)}$ edges, then it contains a rainbow subdivision of any fixed clique $K_t$, with tightness witnessed by the hypercube (Jiang et al., 2021, Wang, 2022).

The generalized rainbow Turán number, $\ex(n, H,\,\text{rainbow-}F)$, for pairs of subgraphs $H, F$, captures the maximum number of rainbow copies of $H$ in a graph with no rainbow $F$. For cycles, sharp asymptotics and extremal constructions exist for all $s, t \geq 4$ (Janzer, 2020, Balogh et al., 2020).

6. Methods, Open Problems, and Rainbow Densities

Methodologically, key tools include reduction to minimal subgraphs, probabilistic and algebraic coloring constructions, weighted counting (e.g. Sidorenko-type inequalities for even cycles), iterative color partition strategies, expansion and regularization arguments, and analytic optimization in the graphon space.

Major open questions and research directions:

• Determining the exact order of growth, and, where possible, the precise constant, of $\ex^*(n, F)$ for general bipartite graphs $F$, especially cycles and trees of larger diameter (Janzer, 2020, Bednar et al., 2022).
• Structural and exact extremal constructions for classes of trees (e.g. for all brooms, caterpillars), and the phase transitions in "k-unique" generalizations.
• Extensions to generalized settings (multiple forbidden/rainbow configurations, system of graphs as color classes), and the full characterization of rainbow Turán densities for non-trees (Im et al., 2023).
• Tightening the gap for cycles between the lower bound $n \log n$ and upper bound $n (\log n)^2$, possibly via new analytic or spectral techniques (Janzer et al., 2022, Wang, 2022).

• "Rainbow Turán number of clique subdivisions" (Jiang et al., 2021)
• "Rainbow Turán Methods for Trees" (Bednar et al., 2022)
• "On the Turán number of the hypercube" (Janzer et al., 2022)
• "On rainbow Turán Densities of Trees" (Im et al., 2023)
• "Rainbow Turán numbers for short brooms" (Byrne et al., 22 Feb 2025)
• "A note on the rainbow Turán number of brooms with length 2 handles" (Halfpap, 17 Dec 2025)
• "On the Rainbow Turán number of paths" (Ergemlidze et al., 2018)
• "The rainbow Turán number of $P_5$" (Halfpap, 2022)
• "Generalized rainbow Turán problems" (Gerbner et al., 2019)
• "Rainbow cycles vs. rainbow paths" (Halfpap et al., 2020)
• "Rainbow Turán problems for paths and forests of stars" (Johnston et al., 2016)
• "Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles" (Janzer, 2020)
• "Rainbow Turán Problem for Even Cycles" (Das et al., 2012)
• "Lower bounds for rainbow Turán numbers of paths and other trees" (Johnston et al., 2019)
• "The generalised rainbow Turán problem for cycles" (Janzer, 2020)