---
title: 'Rainbow Turán Number: Concepts & Results'
url: https://www.emergentmind.com/topics/rainbow-turan-number
type: topic
---

# Rainbow Turán Number: Concepts & Results

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 [2106.13803][2203.13765][2211.02015][2512.15978][1911.06642].

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 \( F \) [2010.14609][2005.08073][1911.06642].

## 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 [2006.01062][1202.3221][2211.02015][2106.13803].

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 [2210.03376][1805.04180][1901.03308][1609.00069].

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 [2512.15978][2502.16057][2203.13765].

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.,
\[
\ex^*(n, C_{2k}) = \Theta(n^{1+1/k})
\]
[2006.01062][2211.02015][2204.08804][2010.14609].

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

The "augmentation" or "reduction" method [2203.13765] 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 [2106.13803]. 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 [2312.15956], 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 \( n \) [1609.00069].

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 [1901.03308][2512.15978][2502.16057].

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),   | [1901.03308], [2210.03376] |
|                           | \( (9k/7+2)\,n \) (upper)              | [1805.04180]             |
| Double Star \( DS_{r,s} \)| \( (s+r-1)\frac{n}{2} \leq ex^* \leq (s+2r)\frac{n}{2} \) | [2203.13765]             |
| Broom \( B_{k,2} \)       | see piecewise formula in [2512.15978]  | [2512.15978]             |
| Cycle \( C_{2k} \)        | \( \Theta(n^{1+1/k}) \)                | [2006.01062], [2211.02015] |

## 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 [2211.02015]. 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 [2106.13803][2204.08804].

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 \( s, t \geq 4 \) [2005.08073][2010.14609].

## 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 [2006.01062][2203.13765].
- 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 [2312.15956].
- 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 [2211.02015][2204.08804].

## 7. References

- "Rainbow Turán number of clique subdivisions" [2106.13803]
- "Rainbow Turán Methods for Trees" [2203.13765]
- "On the Turán number of the hypercube" [2211.02015]
- "On rainbow Turán Densities of Trees" [2312.15956]
- "Rainbow Turán numbers for short brooms" [2502.16057]
- "A note on the rainbow Turán number of brooms with length 2 handles" [2512.15978]
- "On the Rainbow Turán number of paths" [1805.04180]
- "The rainbow Turán number of $P_5$" [2210.03376]
- "Generalized rainbow Turán problems" [1911.06642]
- "Rainbow cycles vs. rainbow paths" [2009.00135]
- "Rainbow Turán problems for paths and forests of stars" [1609.00069]
- "Rainbow Turán number of even cycles, repeated patterns and blow-ups of cycles" [2006.01062]
- "Rainbow Turán Problem for Even Cycles" [1202.3221]
- "Lower bounds for rainbow Turán numbers of paths and other trees" [1901.03308]
- "The generalised rainbow Turán problem for cycles" [2005.08073]

Source: https://www.emergentmind.com/topics/rainbow-turan-number