Forbidding edge-critical graphs as trace in uniform hypergraphs

TL;DR

The paper studies Turán-type extremal problems for forbidding graphs as traces in $r$-uniform hypergraphs, denoted $ex_r(n, Tr_r(G))$. By linking trace-freeness to shadow-freeness and applying shadow stability, it shows that if $G$ is edge-critical with $ icefrac{}{}\chi(G)=s+1$ and $s \ge r \ge 3$, then for large $n$ the unique extremal hypergraph avoiding $Tr_r(G)$ is the complete balanced $s$-partite $r$-graph $T_r(n,s)$, i.e. $ex_r(n, Tr_r(G)) = e(T_r(n,s))$. A crucial step is proving that the shadow of a maximal $Tr_r(G)$-free hypergraph is near-Turán and then upgrading this structure to exact extremality via a light/heavy-edge decomposition and a counting argument, which also yields a characterization of edge-critical graphs in this setting. The results extend prior work on traces (notably for expanded cliques and book graphs) and establish a sharp dichotomy: edge-critical graphs correspond to exact Turán behavior for forbidding their traces in large hypergraphs. The work highlights both the power and the limitations of shadow-based stability in hypergraph trace problems and points to future directions for handling more general trace families.

Abstract

We say a hypergraph $\mathcal{H}$ contains a graph $G$ as trace if there exists a vertex subset $S \subseteq V(\mathcal{H})$ such that $|S| = V(G)$ and $\{e \cap S \mid e \in E(\mathcal{H})\}$ contains $G$ as a subgraph. We use $\mathrm{ex}(n, Tr_r(G))$ to denote the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices not containing $G$ as trace. The study of Turán numbers for traces was initiated by Mubayi and Zhao~(2017) who studied $\mathrm{ex}(n, Tr_r(K_{s+1}))$ where $K_{s+1}$ is a clique on $s+1$ vertices and conjectured the exact value of $\mathrm{ex}(n, Tr_r(K_{s+1}))$. When $r \le s$, the conjecture was covered by a result of Pikhurko~(2013) who gave the exact value of Turán numbers for expanded cliques. Then Gerbner and Picollelli~(2023) gave the exact value for book graphs~($K_{1,1,t}$, the complete tripartite graph with two parts of size one and one part of size $t \ge 2$). We say $G$ is edge-critical if there exists an edge $e \in E(G)$ such that $χ(G - e) < χ(G)$ where $χ(G)$ is the chromatic number of $G$. The definition of edge-critical was given by Simonovits~(1974), who proved that for an edge-critical graph $G$ with $χ(G) = s+1 \ge 3$, the Turán graph $T(n,s)$ is the unique extremal graph for $ex(n,G)$ as $n$ is sufficiently large. In this paper, we further generalize the results of Gerbner and Picollelli~(2023) to edge-critical graphs. More precisely, we prove that for an edge-critical graph $G$ with $χ(G) = s+1$, when $s \ge r \ge 3$ and $n$ is sufficiently large, the $r$-uniform Turán graph $T_r(n,s)$ is the unique extremal hypergraph.

Figures (1)

• Figure 1: The structure in the proof of Claim \ref{['claim: general Ai\' Ai" independent']}. All black edges belong to $\partial \mathcal{H}'$. The red hyperedge $e_0$ correspons to the edge $a_1^{(1)}a_1^{(2)}$ in $\partial \mathcal{H}$.

Theorems & Definitions (15)

• Theorem 1.1: Mubayi and Zhao mubayi2007forbidding
• Conjecture 1.2: Mubayi and Zhao mubayi2007forbidding
• Theorem 1.4: Füredi and Luo FUREDI2023103692
• Theorem 1.5: Gerbner and Picollelli 2023arXiv231005601G
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1: Zykov zykov1949some
• Lemma 2.2: Gerbner and Picollelli 2023arXiv231005601G
• Lemma 2.3: Gerbner and Palmer GERBNER2019103001
• Lemma 2.4: Ma and Qiu MA2020103026