Turán-Type Bounds for Graphs Containing Large $F$-Sparse Sets

Abstract: We study Turán-type extremal problems for graphs containing a large $F$-sparse vertex set, meaning a vertex set whose induced subgraph contains few copies of $F$. For integers $r>s\ge 1$, we prove that if a $K_{r+1}$-free graph $G$ on $n$ vertices contains a set $M$ of size $m\ge \lceil sn/r\rceil$ such that $G[M]$ is $K_{s+1}$-free, then [ e(G)\le m(n-m)+t_s(m)+t_{r-s}(n-m). ] We characterize the equality cases as the complete $r$-partite graphs whose vertex classes split into two balanced groups of total sizes $m$ and $n-m$, consisting of $s$ and $r-s$ classes, respectively. We also prove a color-critical extension for forbidden graphs that embed into a join of two edge-critical graphs, together with an asymptotic extension for general $H$-free graphs in which the prescribed large vertex set spans few copies of a fixed graph $F$ with $χ(F)<χ(H)$.

• The paper establishes exact and asymptotic Turán-type bounds for edge-extremal graphs with large prescribed F-sparse subsets, unifying classical extremal theory.
• It determines optimal edge counts in K₍ᵣ₊₁₎-free graphs via multipartite constructions, extending Turán and Erdős-Stone-Simonovits results.
• The researchers extend these bounds to color-critical graphs, providing precise extremal constructions and tight asymptotic performance.

Turán-Type Extremal Bounds for Graphs with Large $F$-Sparse Sets

Overview

This paper investigates Turán-type extremal graph problems for $H$-free graphs ($H$ typically being a clique or color-critical graph), under the additional condition that there exists a large prescribed subset $M\subseteq V(G)$ whose induced subgraph $G[M]$ contains few (or none) copies of a fixed subgraph $F$. The results extend and unify classical extremal theory such as the Turán and Erdős-Stone-Simonovits theorems, and introduce new bounds and characterization for extremal structures, including exact and asymptotic results depending on the forbidden subgraphs and the structure on $M$.

Main Contributions

Exact Bound for $K_{r+1}$-Free Graphs with Large $K_{s+1}$-Free Sets

The authors determine the extremal number of edges for $K_{r+1}$-free graphs on $H$0 vertices that contain a $H$1-free set $H$2 of size $H$3. The main theorem proves

$H$4

where $H$5 denotes the maximum number of edges in a $H$6-free graph on $H$7 vertices. The extremal graphs are uniquely characterized as complete $H$8-partite graphs where the $H$9 parts are split into two groups forming subgraphs $H$0 and $H$1, with all cross-edges between groups present and parts as balanced as possible.

This structure generalizes Turán's theorem and the case of classical independence number constraints (i.e., the stability number $H$2) to higher clique-forbidding settings. The bounds are sharp and equality characterization is complete.

Extensions to Color-Critical and Double-Edge-Critical Forbidden Graphs

The results are extended to forbidden graphs $H$3 that are double-edge-critical (removal of two vertex-disjoint edges reduces chromatic number by two) and that can be embedded into the join of two edge-critical graphs $H$4. The analysis uses Simonovits' color-critical extremal theorem and a blow-up construction. In this setting, exact and finite-complement bounds are achieved, and the extremal construction remains a natural multipartite generalization, again with a prescribed partition structure dictated by the color classes of the forbidden graph.

The methods also settle the $H$5-progressive-edge-critical case, where forbidden $H$6 reduces chromatic number successively under vertex deletions. This encompasses a broad class of color-critical graphs, further expanding the scope of classical Turán-type extremal bounds.

Asymptotic Generalization for $H$7-Free Graphs with Large $H$8-Sparse Sets

The paper further establishes an asymptotic upper bound for the number of edges in an $H$9-free graph $M\subseteq V(G)$0 on $M\subseteq V(G)$1 vertices containing a set $M\subseteq V(G)$2 of size $M\subseteq V(G)$3 where $M\subseteq V(G)$4 spans at most $M\subseteq V(G)$5 copies of $M\subseteq V(G)$6. If $M\subseteq V(G)$7 and $M\subseteq V(G)$8 are graphs with $M\subseteq V(G)$9, for any fixed $G[M]$0, then for any $G[M]$1, for all sufficiently large $G[M]$2,

$G[M]$3

where the error term is asymptotically negligible. This bound is proved to be tight, with extremal examples provided by multipartite graphs where $G[M]$4- and $G[M]$5-free conditions are realized via suitable partitioning. The proof applies regularity and graph removal lemmas and leverages the structure theory from the earlier exact results.

Strong Claims and Numerical Results

• The paper provides exact (not merely asymptotic) bounds and a uniqueness characterization for extremal graphs, for any $G[M]$6 as long as $G[M]$7.
• The extension to color-critical graphs captures all cases embedded into a join of edge-critical structures, broadening previous results on minimum degree extremal problems.
• The asymptotic result for general graphs demonstrates that the multipartite construction remains extremal as long as the density parameter $G[M]$8 is strictly above the Turán local density threshold, with the error terms parameterized explicitly in terms of $G[M]$9.
• In all nontrivial parameter regimes, the upper bounds are shown to be sharp by explicit construction.

Theoretical Implications

These results systematize the effect of imposed sparse or forbidden subgraphs within large vertex sets on the global edge extremal function, fundamentally relating generalized independence parameters to multipartite extremal constructions. The identification and full characterization of extremal structures under these constraints connect Turán-type phenomena to color-criticality, join-decomposition, and multipartite stability. The extension to the sparse-copy-count case (rather than exact forbiddance) highlights resilience-type behavior and foreshadows further applications in structural graph theory, including stability and saturation phenomena.

Practical Implications and Future Directions

On the practical side, these results guide the design and analysis of extremal examples for property-testing algorithms, combinatorial optimization routines, and network theory, especially where networks of forbidden substructures (cliques or critical motifs) must accommodate large zones of local sparsity or specific forbidden configurations.

Future directions include:

• Development of sharp stability results for near-extremal graphs in the bounded $F$0-copies regime, possibly leveraging stronger regularity and counting lemmas.
• Extension to the case of higher-uniformity hypergraphs and degenerate Turán-type parameters under local forbidden substructure constraints.
• Explicit classification of extremal and near-extremal structures for broader classes of forbidden graphs $F$1, including non-color-critical or non-join-decomposable graphs.

Conclusion

This paper rigorously establishes exact and asymptotic Turán-type upper bounds for edge-extremal graphs under the constraint of prescribed large $F$2-sparse sets, synthesizing and broadening classical extremal graph theory results. The methods extend the reach of traditional multipartite extremal constructions, provide novel equality cases, and open new pathways for studying local-to-global forbidden substructure problems in extremal combinatorics.