Random Turán Problems for Graphs with a Vertex Complete to One Part

Abstract: Given a graph $F$, the random Turán problem asks to determine the maximum number of edges in an $F$-free subgraph of $G_{n,p}$. Prior to this work, the only bipartite graphs $F$ with known tight bounds included certain classes of complete bipartite graphs and theta graphs. We greatly expand upon these examples by proving tight bounds for a number of bipartite graphs which have a vertex complete to one part. We also prove new general upper bounds for this problem which in many cases do significantly better than the only previous known general upper bound due to Jiang and Longbrake. Our proofs utilize dependent random choice together with the recent technique of balanced vertex supersaturation in conjunction with hypergraph containers.

• The paper presents novel sharp asymptotics for random Turán numbers, expanding known results for bipartite graphs with a vertex complete to one part.
• It employs dependent random choice and hypergraph container methods to derive improved upper bounds, tightening previous exponent gaps.
• The study identifies clear threshold regimes that match conjectured behaviors, advancing both structural extremal theory and random graph analysis.

Random Turán Numbers for Bipartite Graphs with a Vertex Complete to One Part

Introduction and Context

The paper "Random Turán Problems for Graphs with a Vertex Complete to One Part" (2604.02264) advances the understanding of random Turán-type extremal problems in sparse random graphs $G_{n,p}$, focusing on bipartite forbidden subgraphs $F$ that exhibit a vertex adjacent to an entire side of their bipartition. The main focus is to determine, up to asymptotics, the maximum number of edges in an $F$-free subgraph of $G_{n,p}$ (the random Turán number $ex(G_{n,p},F)$) in regimes where $F$ is bipartite and $p$ is subconstant.

Previous work yielded tight random Turán asymptotics for non-bipartite $F$—notably by Conlon-Gowers and Schacht—and, for bipartite $F$, essentially only for large complete bipartite graphs and specific classes of theta graphs. Theoretical tools such as dependent random choice (DRC), hypergraph containers, and the framework of supersaturation have enabled recent progress on extremal and random problems, but systematic results for broad bipartite classes have lagged due to complications arising from degenerate (bipartite) cases and connections to hard open conjectures, such as Sidorenko's.

Technical Innovations and Main Results

Sidorenko Graphs and Structured Bipartite Graphs

The authors leverage the fact that bipartite graphs where one vertex is complete to an entire part are Sidorenko, thanks to Conlon-Fox-Sudakov, justifying the focus on these classes. The present work establishes:

• Tight upper and lower bounds for $ex(G_{n,p},F)$ for a broad infinite family of bipartite graphs $F$0 that have a vertex complete to one part.
• A new suite of general upper bounds (sometimes significantly improving upon Jiang-Longbrake) for random Turán numbers for this class.
• A unified and more general framework (including parameters $F$1 defined via technical density and balancedness conditions) for random Turán numbers of $F$2 with restricted degree structures.

The main technical innovation is the deployment of a vertex balanced supersaturation approach, allowing for the translation of extremal (deterministic) arguments with DRC into the probabilistic setting, followed by refined use of the container method to enumerate $F$3-free subgraphs with high concentration.

Strong Asymptotics and Explicit Ranges

General Upper Bound:

If $F$4 is bipartite, admits a bipartition $F$5, and possesses a vertex $F$6 complete to $F$7, and all other $F$8 have degree $F$9, then:

• For all $F$0,
• $F$1
• These bounds are tight when $F$2 contains an appropriate $F$3.

This result is strictly sharper than the best previous general bound [Jiang-Longbrake] in several parameter regimes.

Tight Families:

• For graphs $F$4 formed from a multigraph $F$5 by subdividing each edge and joining a vertex $F$6 to all of $F$7, the paper tightly determines $F$8 for all $F$9.
• This analysis extends using a general result for $G_{n,p}$0-semi-bounded graphs, covering all graphs with a "dominating vertex" (as above) and all other vertices in $G_{n,p}$1 having degree exactly $G_{n,p}$2.
• Asymptotics for the three standard random Turán regimes (sparse, intermediate, and dense) are explicitly described; these include a flat "middle range" where $G_{n,p}$3 is (up to logs) $G_{n,p}$4 and does not depend on $G_{n,p}$5.

Balanced Supersaturation:

Theoretical contributions include a general vertex balanced supersaturation result: for $G_{n,p}$6-semi-bounded $G_{n,p}$7, any sufficiently dense graph contains a large collection of copies of $G_{n,p}$8 such that no small vertex set hosts too many copies, extending and refining prior supersaturation tools in random Turán analysis.

Comparison with Prior Work

The only previous tight families for bipartite $G_{n,p}$9 were (large) complete bipartite graphs [Morris-Saxton] and some theta graphs [McKinley-Spiro]. This work vastly enlarges the family for which tight random Turán asymptotics are established. Further, the explicit identification of the middle range and the handling of arbitrary edge-multiplicity structures (via subdivided multigraphs) marks a significant departure in achievable explicitness and generality.

In addition, the upper bounds are proved via an overview of DRC, vertex (and consequently edge) supersaturation via containers, and a detailed analysis of 2-density and "semi-boundedness" parameters.

Substantial Claims and Contradictions

Two major bold claims are highlighted:

• For a broad class of bipartite $ex(G_{n,p},F)$0, including non-complete and non-theta graphs, the three-regime random Turán behavior conjectured by McKinley-Spiro holds (with precise thresholds and exponents).
• For these families, the upper bounds obtained are strictly sharper than previously known general bounds and are tight in all ranges of $ex(G_{n,p},F)$1.

No contradictory claims are made relative to previous literature; rather, the results substantially strengthen and extend prior knowledge.

Numerical Strength and Implications

The theorems explicitly yield bounds of the following forms:

$ex(G_{n,p},F)$2 Regime	$ex(G_{n,p},F)$3
$ex(G_{n,p},F)$4	$ex(G_{n,p},F)$5
$ex(G_{n,p},F)$6	$ex(G_{n,p},F)$7
$ex(G_{n,p},F)$8	$ex(G_{n,p},F)$9

where the exponents are expressed in terms of parameters of $F$0 (such as $F$1, maximum/minimum degrees, and the structure of $F$2 for subdivided multigraphs). The explicit identification of the width of the flat $F$3-independent regime is notable.

From a methodological perspective, these sharp results in the degenerate, bipartite setting facilitate further progress in Sidorenko-type conjectures for random graphs, and the paper rigorously clarifies the delicate combinatorial structure required for the validity of the conjectured random Turán formula.

Theoretical and Practical Implications

Theoretical

• Provides a template for addressing random Turán problems in other wide and structurally intricate bipartite graphs, deepening understanding of "degenerate" extremal graph theory in random environments.
• Confirms, for semi-bounded families, the McKinley-Spiro conjecture's prediction of the threshold demarcation in $F$4 and existence of a flat regime.
• Connects random Turán thresholds and graph density parameters directly to Sidorenko-type questions, reinforcing the probabilistic interpretation of the Sidorenko property.
• Naturally extends to conjectured wider classes (i.e., multi-dominating-vertex bipartite graphs), with explicit formulation of possible further generalizations.

Practical

While primarily theoretical, these asymptotic random Turán bounds give, in principle, quantitative guidance for problems such as subgraph-freeness in randomly sampled communication, dependency, or interaction graphs with forbidden bipartite "hub" structures—important in network theory, sparse matrix theory, and extremal combinatorics settings.

Prospects for Future Research

Obvious future directions include:

• Extension to $F$5-bounded graphs with $F$6 (i.e., more than one dominating vertex).
• Systematic identification and analysis of minimal forbidden subgraphs admitting tight random Turán results via the supersaturation/containers paradigm.
• Further exploration of the relationship between random Turán numbers, Sidorenko’s conjecture, and related density inequalities.
• Application of vertex-balanced supersaturation techniques to other extremal random combinatorial structures, including uniform hypergraphs.

Conclusion

This work provides a comprehensive, technically robust extension of the random Turán theory for structurally rich bipartite graphs with a dominating vertex, integrating advanced probabilistic and combinatorial techniques. The resulting theorems fill a significant gap in the random Turán landscape, expanding the classes of graphs with tight extremal and probabilistic thresholds, and clarify the interaction between Sidorenko-type properties and probabilistic extremal graph theory.