Subgraphs with large minimum $\ell$-degree in hypergraphs where almost all $\ell$-degrees are large

Let $G$ be an $r$-uniform hypergraph on $n$ vertices such that all but at most $\varepsilon \binom{n}{\ell}$ $\ell$-subsets of vertices have degree at least $p \binom{n-\ell}{r-\ell}$. We show that $G$ contains a large subgraph with high minimum $\ell$-degree.


Introduction
Given r ∈ N and a set A, we write A (r) for the collection of all r-subsets of A and [n] for the set {1, 2, . . . n}. An r-graph, or r-uniform hypergraph, is a pair G = (V, E), where V = V (G) is a set of vertices and E = E(G) ⊆ V (r) is a collection of r-subsets, which constitute the edges of G. We say G is nonempty if it contains at least one edge and set v(G) = |V (G)| and e(G) = |E(G)|. A subgraph of G is an r-graph H with V (H) ⊆ V (G) and E(H) ⊆ E(G). The subgraph of G induced by a set X ⊆ V (G) is G[X] = (X, E(G) ∩ X (r) ).
Let F be a family of nonempty r-graphs. If G does not contain a copy of a member of F as a subgraph, we say that G is F -free. The Turán number ex(n, F ) of a family F is the maximum number of edges in an F -free r-graph on n vertices, and its Turán density is the limit π(F ) = lim n→∞ ex(n, F )/ n r (this is easily shown to exist). Let K (r) t = ([t], [t] (r) ) denote the complete r-graph on t vertices. Determining π(K (r) t ) for any t > r ≥ 3 is a major problem in extremal combinatorics. Turán [19] famously conjectured in 1941 that π(K (3) 4 ) = 5/9, and despite much research effort this remains open [8]. In this paper we shall be interested in some variants of Turán density.
The neighbourhood N(S) of an ℓ-subset S ∈ V (G) (ℓ) is the collection of (r − ℓ)-subsets T ∈ V (G) (r−ℓ) such that S ∪ T is an edge of G. The degree of S is the number deg(S) of edges of G containing S, that is, deg(S) = |N(S)|. The minimum ℓ-degree of G, δ ℓ (G), is defined to be the minimum of deg(S) over all ℓ-subsets S ∈ V (G) (ℓ) . The Turán ℓ-degree threshold ex ℓ (n, F ) of a family F of r-graphs is the maximum of δ ℓ (G) over all F -free r-graphs G on n vertices. It can be shown [11,9] that the limit π ℓ (F ) = lim n→∞ ex ℓ (n, F )/ n−ℓ r−ℓ exists; this quantity is known as the Turán ℓ-degree density of F . A simple averaging argument shows that 0 ≤ π r−1 (F ) ≤ · · · ≤ π 2 (F ) ≤ π 1 (F ) = π(F ) ≤ 1, and it is known that π ℓ (F ) = π(F ) in general (for ℓ / ∈ {0, 1}). In the special case where (r, ℓ) = (r, r − 1), π r−1 (F ) is known as the codegree density of F .
there is a constant p ∈ [0, 1] and a sequence of nonnegative reals ε n → 0 as n → ∞ such that We refer to the supremum of all p ≥ 0 for which (ii) is satisfied as the density of the sequence G and denote it by ρ(G) .
We can define the analogue of Turán density for (r, ℓ)-sequences.
Our main result show that every large r-graph G contains a 'somewhat large' subgraph H with minimum ℓ-degree satisfying This immediate implies the π ⋆ ℓ (F ) = π ℓ (F ) for all families F of r-graphs.
We note that the (tight) upper bounds for codegree densities π 2 (F ) for 3-graphs F obtained by flag algebraic methods in [5,6,7] actually relied on giving upper bounds for π ⋆ ℓ (F ). Corollary 4 provides theoretical justification for why this strategy could give optimal bounds.
Consider a (p, ε, (1, 2))-quasirandom 3-graph G for some p > 4 √ ε > 0. As noted above, δ However, as we show below, we cannot guarantee the existence of any subgraph with strictly positive codegree on more than 2/ε + 1 vertices: our lower bound on m above in terms of an inverse power of the error parameter ε is thus sharp up to the value of the exponent. Proposition 7. For every p ∈ (0, 1) and every ε > 0, there exists n 0 such that for all n ≥ n 0 there exist (p, 2ε, (1, 2))-quasirandom 3-graphs in which every subgraph on m ≥ ⌊ε −1 ⌋ + 1 vertices has minimum codegree equal to zero. a (p, ε, (1, 2))-quasirandom 3-graph on n vertices. Such a 3-graph can be obtained for example by taking a typical instance of an Erdős-Rényi random 3-graph with edge probability p. Consider a balanced partition of V into N = ⌊ε −1 ⌋ sets V = N i=1 V i with ⌊n/N⌋ ≤ |V 1 | ≤ |V 2 | ≤ . . . ≤ |V N | ≤ ⌈n/N⌉. Now let G ′ be the 3-graph obtained from G by deleting all triples that meet some V i in at least two vertices for some i: By construction, every set of N + 1 vertices in G ′ must contain at least two vertices from the same V i , and thus must induce a subgraph of G ′ with minimum codegree zero. Note that e(G) − e(G ′ ) ≤ Nn ⌈n/N ⌉ 2 ≤ n 3 /N ≤ εn 3 . Since G is (p, ε, (1, 2))-quasirandom, it follows that G ′ is (p, 2ε, (1, 2))-quasirandom.
2 Finding high minimum ℓ-degree subgraphs in r-graphs with large δ ε ℓ In this section we show how we can extract arbitrarily large subgraphs with high minimum ℓ-degree from sufficiently large r-graphs with sufficiently small error ε. To do so, we will need Azuma's inequality (see e.g. [1]).
Call an ℓ-subset S ∈ V (ℓ) poor if deg(S) < p n−ℓ r−ℓ , and rich otherwise. Let P be the collection of all poor ℓ-subsets. By our assumption on δ ε ℓ (G), |P| ≤ ε n ℓ . As each poor ℓ-subset is contained in n−ℓ m−ℓ m-subsets, it follows that there are at least m-subsets of vertices which do not contain any poor ℓ-subsets.
Given an ℓ-subset S ∈ V (ℓ) \ P, we call an m-subset T of V bad for S if S ⊆ T and N(S) ∩ T (r−ℓ) ≤ (p − δ) m−ℓ r−ℓ . Let φ S be the number of bad m-subsets for S. We claim that Observe that Let X be the random variable N(S) ∩ T (r−ℓ) , where T is an (m − ℓ)-subset of V \ S picked uniformly at random. We consider the vertex exposure martingale on T . Let Z i be the ith exposed vertex in T . Define X i = E(X|Z 1 , . . . , Z i ). Note that {X i : i = 0, 1, . . . , m − ℓ} is a martingale and X 0 ≥ p m−ℓ r−ℓ . Moreover, |X i − X i−1 | ≤ m−ℓ−1 r−ℓ−1 < m−1 r−ℓ−1 . Thus, by Lemma 8 applied with λ = δ m r−ℓ and c i = m−1 r−ℓ−1 , we have Hence (3) holds.
An m-subset T of V is called bad if it is bad for some S ∈ V (ℓ) \P. The number of bad m-subsets is at most where the last three inequalities hold by our choice of m 0 , by inequality (1), and by our assumption on δ, respectively. Together with (2), this shows there exists an m-subset inside which there is no poor ℓ-subsets and in which every rich ℓ-subset has degree at least (p − δ) m−ℓ r−ℓ . Such a set clearly gives us an induced subgraph of G on m vertices with minimum ℓ-degree at least (p − δ) m−ℓ r−ℓ .
Even more: can one always extract subgraphs with large minimum codegree from (1, 1, 1)-quasirandom graphs? Even obtaining large subgraphs with non-zero minimum codegree remains an open problem for this weaker notion of quasirandomness.