---
title: Trace-Turán Numbers
url: https://www.emergentmind.com/topics/trace-turan-numbers
type: topic
---

# Trace-Turán Numbers

A trace-Turán number describes, for a given graph $G$, the maximum number of edges possible in an $r$-uniform hypergraph on $n$ vertices that avoids containing $G$ as a trace. A trace is defined by the projection of hyperedges to a suitably chosen vertex subset, recovering the original graph $G$ as a (sub)graph within these projections. Trace-Turán numbers generalize classical Turán-type extremal functions from ordinary graphs to the combinatorially richer realm of uniform hypergraphs. Their study connects classical extremal combinatorics, hypergraph theory, design theory, and embedding problems.

## 1. Definitions and Concepts

Let $G$ be a simple graph, and let $\mathcal{H}$ be an $r$-uniform hypergraph with vertex set $V(\mathcal{H})$. The hypergraph $\mathcal{H}$ contains $G$ as a trace if there exists a subset $S \subseteq V(\mathcal{H})$, $|S| = |V(G)|$, such that the 2-shadow defined by $\{e \cap S : e \in E(\mathcal{H})\}$ contains $G$ as a subgraph.

The trace-Turán number is denoted by
$$
\mathrm{ex}(n, \mathrm{Tr}_r(G))
$$
and is defined as the maximum number of edges in an $n$-vertex $r$-uniform hypergraph that does not contain $G$ as a trace. This notion is equivalent to the induced Berge-$G$ extremal problem, with a $G$-trace corresponding to an induced Berge-$G$ in $\mathcal{H}$.

Key related constructions include:

- The $r$-uniform Turán graph $T_r(n, s)$: the complete balanced $s$-partite $r$-graph, where vertex set $[n]$ is partitioned into $s$ classes as equally as possible, and all $r$-sets that meet each class in at most one vertex form the edge set. For $r=2$, this recovers the classical Turán graph $T(n, s)$.
- The notion of edge-criticality: a graph $G$ is edge-critical if it has an edge $e$ for which $\chi(G - e) < \chi(G)$, where $\chi(G)$ is the chromatic number.

## 2. Principal Theorems and Classification

For edge-critical graphs $G$ with $\chi(G) = s+1$, it is established that, for all integers $r$ satisfying $3 \le r \le s$ and all sufficiently large $n$,
$$
\mathrm{ex}(n, \mathrm{Tr}_r(G)) = e(T_r(n, s)) = N(K_r, T(n,s))
$$
where $N(K_r, T(n,s))$ counts the number of $r$-cliques in $T(n,s)$. Moreover, $T_r(n, s)$ is the unique extremal hypergraph, generalizing the classical extremal characterization for graphs due to Simonovits [2601.09500].

Special cases:

- $G = K_{s+1}$ (clique): recovers the Mubayi–Zhao conjecture for $r \leq s$, proved by Pikhurko.
- $G = B^t_2 = K_{1,1,t}$ (book graph): recovers the result that for $r=3$,
  $$
  \mathrm{ex}(n, \mathrm{Tr}_3(B^t_2)) = \lfloor (n-1)^2/4 \rfloor
  $$
  as determined by Gerbner–Picollelli.

For general $F$ and uniformity $r$, Füredi and Luo [2002.07350] established an asymptotic upper bound:
$$
\mathrm{ex}_r(n, B_{ind} F) = O\Big(\max_{2 \leq s \leq r} \mathrm{ex}(n, K_s, F)\Big)
$$
where $\mathrm{ex}(n, K_s, F)$ is the maximal number of $K_s$ in an $F$-free graph. For outerplanar graphs $F$ (the class $G_{tri}$), as $n \to \infty$,
$$
\mathrm{ex}_r(n, B_{ind} F) = O(\mathrm{ex}(n, F))
$$
with analogous results for cycles and forests.

## 3. Structural Proofs and Extremal Constructions

The main extremal proofs employ several key techniques:

- **Light–Heavy Edge Decomposition:** Iteratively remove "light" edges (those containing an $(r-1)$-subset with low degree) to leave only "heavy" edges, reducing the problem to a structure reminiscent of a Turán graph with controllable perturbations [2601.09500].
- **Shadow Graph Stability:** The $2$-shadow (graph formed from pairs in edges of the hypergraph) of the heavy component preserves $G$-freeness and nearly maximizes the count of $K_r$ subgraphs. Application of graph stability theorems (e.g., Ma–Qiu's extension of Simonovits' theorem) ensures proximity to $T(n, s)$.
- **Exceptional Set Elimination:** A stability partition is refined to show the negligible size of any exceptional class, guaranteeing that the shadow is strictly $T(n,s)$, hence every hyperedge in the hypergraph corresponds precisely to a $K_r$ in the Turán graph.
- **Edge-Criticality Mechanism:** The chromatic property ensures that if an additional edge appears within a color class, one can build a trace of $G$ within the hypergraph.

For stars and book graphs, construction proceeds by careful edge deletion in balanced clique partitions guided by covering design theory, ensuring no trace of $K_{1,t}$ arises, with many cases featuring covering-based bounds that are provably tight [2206.05884].

## 4. Special Cases, Extensions, and Comparison of Bounds

Trace-Turán numbers recover or strengthen a variety of classical results in extremal graph theory:

- For fixed graphs $F$ and $r < \chi(F)$, all generalized functions $\mathrm{ex}(n, F)$, $\mathrm{ex}_r(n, BF)$, $\mathrm{ex}_r(n, B_{ind} F)$ are $\Theta(n^r)$, achieved by the complete $(\chi(F)-1)$-partite $r$-graph.
- If $r \geq |V(F)|$, then $\mathrm{ex}(n, K_r, F) = 0$ and the induced Berge problem may still attain $\Theta(n^r)$ (e.g., $F = K_{2,2}$).
- For bipartite $F$, bounds for $\mathrm{ex}(n, K_r, F)$ are not well understood for $r \geq 3$.
- For non-bipartite $F$, a uniformity threshold can force $\mathrm{ex}_r(n, BF) = o(n^2)$, but the exact threshold and induced case remain open [2002.07350].

Improved lower bounds for stars $K_{1,t}$ involve minimal covering designs: removing the minimal set of $r$-sets (blocks) covering all $(r-1)$-subsets in each clique partition. This leads to sharp bounds under specified divisibility constraints and covering existence (e.g., with Steiner systems).

Upper bounds for $K_{2,t}$ traces in 3-uniform hypergraphs use fine-grained co-degree analysis, bounding the number of edges with high pairwise codegree, and Ramsey-type lemmas to control neighborhood intersection structure. These yield precise asymptotics, particularly with small $t$ [2206.05884].

## 5. Methodological and Theoretical Tools

The following methodologies and results are central:

- **Design Theory and Covering Designs:** Used to construct extremal cases without traces of stars, exploiting combinatorial coverings such as Steiner systems.
- **a-Core Partitioning:** Partitioning an $r$-partite hypergraph into a core with high $(r-1)$-degrees and a remainder, facilitating inductive bounds [2002.07350].
- **Strongly Representable Families:** Forbidding certain intersection patterns in link systems to avoid traces of star graphs.
- **Double Counting and Co-Degree Partitioning:** For bounding edge numbers in hypergraphs avoiding certain trace-graphs.
- **Stability Theorems and Expansion Arguments:** To eliminate exceptional sets and ensure unique extremal structures.
- **Shadow Graph Analysis:** Translating properties of the original hypergraph to its lower-dimensional "shadow," often reducing trace problems to classical graph extremal problems.

## 6. Open Questions and Directions

The following unresolved problems remain central:

- Determining the exact order of magnitude for $\mathrm{ex}_r(n, B_{ind} F)$ in the regime $\chi(F) < r < |V(F)|$ where $\mathrm{ex}(n, K_r, F)$ is still undetermined.
- Characterizing all graphs $F$ for which $\mathrm{ex}_r(n, B_{ind} F) = \Theta(\mathrm{ex}(n, F))$ for every fixed $r$ (the outerplanar family $G_{tri}$ is one confirmed case).
- Sharpening the constant factors in the general upper bound relating trace-Turán numbers to generalized Turán numbers.
- Determining uniformity thresholds $r_0(F)$ above which trace-Turán numbers drop below quadratic growth.
- Establishing exact asymptotics for specific small graphs and uniformities, particularly for bipartite $F$ and low $t$ in $K_{2,t}$-trace problems [2002.07350, 2206.05884].

## 7. Summary Table: Key Trace-Turán Results

| Graph $G$            | Uniformity $r$    | $\mathrm{ex}(n, \mathrm{Tr}_r(G))$                    | Extremal Construction                          |
|----------------------|-------------------|------------------------------------------------------|-----------------------------------------------|
| $K_{s+1}$            | $r \leq s$        | $e(T_r(n,s))$                                        | Complete balanced $s$-partite $r$-graph       |
| Edge-critical $G$    | $3 \leq r \leq s$ | $e(T_r(n,s))$                                        | Unique: $T_r(n,s)$                            |
| Book $K_{1,1,t}$     | $r=3$             | $\lfloor (n-1)^2/4\rfloor$                           | Bipartite graph construction                  |
| Star $K_{1,t}$       | $r=3$             | $\frac{n}{6}(t^2-2)$ or $\frac{n}{6}(t^2-1)$ (exact) | Covering design construction                  |
| $K_{2,t}$-trace      | $r=3$             | $O\left(n^{3/2}\right)$, explicit constant           | Codegree partition, Ramsey lemma              |

These results situate trace-Turán theory as a robust and generative extension of classical Turán extremal problems, blending combinatorial, probabilistic, and design-theoretic methods and motivating further structural exploration—especially concerning non-edge-critical and bipartite forbidden graphs [2601.09500, 2206.05884, 2002.07350].

Source: https://www.emergentmind.com/topics/trace-turan-numbers