Trace-Turán Numbers

• Trace-Turán numbers are extremal functions defined for r-uniform hypergraphs that avoid a given graph as a trace, thereby generalizing classical Turán numbers.
• Key results show that, for edge-critical graphs like cliques and book graphs, unique extremal constructions such as balanced Turán graphs emerge.
• Methodologies such as light–heavy edge decomposition and shadow graph analysis connect graph stability and design theory with hypergraph extremal problems.

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 field 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 (Wang et al., 14 Jan 2026).

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 (Furedi et al., 2020) 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 (Wang et al., 14 Jan 2026).
• 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 (Qian et al., 2022).

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 (Furedi et al., 2020).

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 $t$ (Qian et al., 2022).

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 (Furedi et al., 2020).
• 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 (Furedi et al., 2020, Qian et al., 2022).

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 (Wang et al., 14 Jan 2026, Qian et al., 2022, Furedi et al., 2020).