REVIEW 2 cited by
On the boundedness of degenerate hypergraphs
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
abstract
We investigate the impact of a high-degree vertex in Tur\'{a}n problems for degenerate hypergraphs (including graphs). We say an $r$-graph $F$ is bounded if there exist constants $\alpha, \beta>0$ such that for large $n$, every $n$-vertex $F$-free $r$-graph with a vertex of degree at least $\alpha \binom{n-1}{r-1}$ has fewer than $(1-\beta) \cdot \mathrm{ex}(n,F)$ edges. The boundedness property is crucial for recent works~\cite{HHLLYZ23a,DHLY24} that aim to extend the classical Hajnal--Szemer\'{e}di Theorem and the anti-Ramsey theorems of Erd\H{o}s--Simonovits--S\'{o}s. We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for $3$-graphs, strengthening a theorem by Kostochka--Mubayi--Verstra\"{e}te~\cite{KMV15}.
Forward citations
Cited by 2 Pith papers
-
Tiling $H$ in dense graphs
The asymptotic maximum number of edges in a graph with H-matching number below beta n is determined for the H-shaped tree, refuting Lang's conjecture.
-
Density Hajnal--Szemer\'{e}di theorem for cliques of size four
For large n and any k ≤ n/4, the maximum number of edges in an n-vertex graph with no k+1 disjoint K4's is asymptotically Ξ(n,k), a piecewise quadratic with five regimes.
Discussion (0). Continue with ORCID to comment.