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1 This is the published versio of a paper published i The Electroic Joural of Combiatorics. Citatio for the origial published paper (versio of record: Falgas-Ravry, V., Lo, A. (2018 Subgraphs with large miimum l-degree i hypergraphs where almost all l-degrees are large The Electroic Joural of Combiatorics, 25(2: P2.18 Access to the published versio may require subscriptio. N.B. Whe citig this work, cite the origial published paper. Permaet lik to this versio:

2 Subgraphs with large miimum l-degree i hypergraphs where almost all l-degrees are large Victor Falgas-Ravry Istitutioe för matematik och matematisk statistik Umeå Uiversitet Umeå, Swede victor.falgas-ravry@umu.se Alla Lo Departmet of Mathematics Uiversity of Birmigham Birmigham, Uited Kigdom s.a.lo@bham.ac.uk Submitted: Oct 20, 2016; Accepted: Feb 23, 2018; Published: Apr 27, 2018 c The authors. Released uder the CC BY-ND licese (Iteratioal 4.0. Abstract Let G be a r-uiform hypergraph o vertices such that all but at most ε ( l l-subsets of vertices have degree at least p ( l. We show that G cotais a large subgraph with high miimum l-degree. Keywords: r-uiform hypergraphs, l-degree, extremal hypergraph theory Mathematics Subject Classificatios: 05C65, 05D99 1 Itroductio Give r N ad a set A, we write A (r for the collectio of all r-subsets of A ad [] for the set {1, 2,... }. A r-graph, or r-uiform hypergraph, is a pair G = (V, E, where V = V (G is a set of vertices ad E = E(G V (r is a collectio of r-subsets, which costitute the edges of G. We say G is oempty if it cotais at least oe edge ad set v(g = V (G ad e(g = E(G. A subgraph of G is a r-graph H with V (H V (G ad E(H E(G. The subgraph of G iduced by a set X V (G is G[X] = (X, E(G X (r. Let F be a family of oempty r-graphs. If G does ot cotai a copy of a member of F as a subgraph, we say that G is F-free. The Turá umber ex(, F of a family F is the maximum umber of edges i a F-free r-graph o vertices, ad its Turá desity is the limit π(f = lim ex(, F/ ( (r r (this is easily show to exist. Let K t = ([t], [t] (r deote the complete r-graph o t vertices. Determiig π(k (r t for ay t > r 3 is a Supported by VR startig grat Supported by EPSRC first grat EP/P002420/1 the electroic joural of combiatorics 25(2 (2018, #P2.18 1

3 major problem i extremal combiatorics. Turá [19] famously cojectured i 1941 that π(k (3 4 = 5/9, ad despite much research effort this remais ope [8]. I this paper we shall be iterested i some variats of Turá desity. The eighbourhood N(S of a l-subset S V (G (l is the collectio of (r l-subsets T V (G ( such that S T is a edge of G. The degree of S is the umber deg(s of edges of G cotaiig S, that is, deg(s = N(S. The miimum l-degree of G, δ l (G, is defied to be the miimum of deg(s over all l-subsets S V (G (l. The Turá l-degree threshold ex l (, F of a family F of r-graphs is the maximum of δ l (G over all F-free r-graphs G o vertices. It ca be show [11, 9] that the limit π l (F = lim ex l (, F/ ( l exists; this quatity is kow as the Turá l-degree desity of F. A simple averagig argumet shows that 0 π r 1 (F... π 2 (F π 1 (F = π(f 1, ad it is kow that π l (F π(f i geeral (for l / {0, 1}. I the special case where (r, l = (r, r 1, π r 1 (F is kow as the codegree desity of F. There has bee much research o Turá l-degree threshold for r-graphs whe (r, l = (3, 2. I the late 1990s, Nagle [12] ad Nagle ad Czygriow [2] cojectured that π 2 (K (3 4 = 1/4 ad π 2 (K (3 4 = 1/2, respectively. Here K (3 4 deotes the 3-graph obtaied by removig oe edge from K (3 4. Falgas-Ravry, Pikhurko, Vaugha ad Volec [6, 7] recetly proved π 2 (K (3 4 = 1/4, settlig the cojecture of Nagle, ad showed all earextremal costructios are close (i edit distace to a set of quasiradom touramet costructios of Erdős ad Hajal [3]. The lower boud π 2 (K (3 4 1/2 also comes from a quasiradom costructio, which is due to Rödl [17]. For t > r 3, the codegree desity π r 1 (K (r t has bee studied by Falgas-Ravry [4], Lo ad Markström [9] ad Sidoreko [18]. Recetly, Lo ad Zhao [10] showed that 1 π r 1 (K (r t = Θ(l t/t r 1 for r 3. Oe variat of l-degree Turá desity is to study r-graphs i which almost all l-subsets have large degree. To be precise, give ε > 0, let δl ε (G be the largest iteger d such that all but at most ε ( v(g l of the l-subsets S V (G (l satisfy deg(s d. Note that r-graphs with large δl ε(g but with small δ l(g arise aturally. For istace, the reduced graphs R obtaied from r-graphs with large miimum l-degree after a applicatio of hypergraph regularity lemma have large δl ε(r. Defiitio 1 ((r, l-sequece. Let 1 l < r. We say that a sequece G = (G N of r-graphs is a (r, l-sequece if (i v(g as ad (ii there is a costat p [0, 1] ad a sequece of oegative reals ε 0 as such that δ ε l (G p ( v(g l for each. We refer to the supremum of all p 0 for which (ii is satisfied as the desity of the sequece G ad deote it by ρ(g. We ca defie the aalogue of Turá desity for (r, l-sequeces. the electroic joural of combiatorics 25(2 (2018, #P2.18 2

4 Defiitio 2. Let 1 l < r. Let F be a family of oempty r-graphs. Defie { } πl (F := sup ρ(g : G is a (r, l-sequece of F-free r-graphs. Our mai result show that every large r-graph G cotais a somewhat large subgraph H with miimum l-degree satisfyig δ l (H/ ( v(h l δ ε l (G/ ( v(g l. Here somewhat large meas v(h = Ω(ε 1/l. Theorem 3. Let 1 l < r. For ay fixed δ > 0, there exists m 0 > 0 such that ay r-graph G o m m 0 vertices with δl ε(g p( l for some ε m l /2 cotais a iduced subgraph H o m vertices with ( m l δ l (H (p δ. r l This immediate implies the π l (F = π l(f for all families F of r-graphs. Corollary 4. For ay 1 l < r ad ay family F of oempty r-graphs, π l (F = π l(f. We ote that the (tight upper bouds for codegree desities π 2 (F for 3-graphs F obtaied by flag algebraic methods i [5, 6, 7] actually relied o givig upper bouds for πl (F. Corollary 4 provides theoretical justificatio for why this strategy could give optimal bouds. 1.1 Quasiradomess i 3-graphs Oe of the mai motivatios for this ote comes from recet work of Reiher, Rödl ad Schacht [13, 14, 15, 16] o extremal questios for quasiradom hypergraphs. These authors studied the followig otio of quasiradomess for 3-graphs. Defiitio 5 ((1,2-quasiradomess. A 3-graph G is (p, ε, (1, 2-quasiradom if for every set of vertices X V ad every set of pairs of vertices P V (2, the umber e 1,2 (X, P of pairs (x, uv X P such that {x} {uv} E(G satisfies: e 1,2 (X, P p X P εv(g 3. We defie a (1, 2-quasiradom sequece ad the correspodig extremal desity, deoted by π (1,2 qr (F, aalogously to the way we defied (r, l-sequeces ad πl (F i Defiitios 1 ad 2. It is ot difficult to see that π (1,2 qr (F π(f for all families F of 3- graphs. Moreover, a (p, ε, (1, 2-quasiradom 3-graph G satisfies δ ε 2 (G (p 4 εv(g. Hece, Theorem 3 ad Corollary 6 imply the followig. Corollary 6. For ay family of oempty 3-graphs F, π (1,2 qr (F π 2 (F. the electroic joural of combiatorics 25(2 (2018, #P2.18 3

5 Cosider a (p, ε, (1, 2-quasiradom 3-graph G for some p > 4 ε > 0. As oted above, δ ε 2 (G (p 4 εv(g. Thus provided v(g is sufficietly large, Theorem 3 tells us we ca fid a subgraph H of G o m = Ω(ε 1/4 vertices with strictly positive miimum codegree (at least (p 4 εm. However, as we show below, we caot guaratee the existece of ay subgraph with strictly positive codegree o more tha 2/ε + 1 vertices: our lower boud o m above i terms of a iverse power of the error parameter ε is thus sharp up to the value of the expoet. Propositio 7. For every p (0, 1 ad every ε > 0, there exists 0 such that for all 0 there exist (p, 2ε, (1, 2-quasiradom 3-graphs i which every subgraph o m ε vertices has miimum codegree equal to zero. Proof. Let G = (V, E be a (p, ε, (1, 2-quasiradom 3-graph o vertices. Such a 3- graph ca be obtaied for example by takig a typical istace of a Erdős Réyi radom 3-graph with edge probability p. Cosider a balaced partitio of V ito N = ε 1 sets V = N i=1 V i with /N V 1 V 2... V N /N. Now let G be the 3-graph obtaied from G by deletig all triples that meet some V i i at least two vertices for some i: 1 i N. By costructio, every set of N + 1 vertices i G must cotai at least two vertices from the same V i, ad thus must iduce a subgraph of G with miimum codegree zero. Note that e(g e(g N ( /N 2 3 /N ε 3. Sice G is (p, ε, (1, 2-quasiradom, it follows that G is (p, 2ε, (1, 2-quasiradom. 2 Fidig high miimum l-degree subgraphs i r-graphs with large δ ε l I this sectio we show how we ca extract arbitrarily large subgraphs with high miimum l-degree from sufficietly large r-graphs with sufficietly small error ε. To do so, we will eed Azuma s iequality (see e.g. [1]. Lemma 8 (Azuma s iequality. Let {X i : i = 0, 1,... } be a martigale with X i X i 1 c i for all i. The for all positive itegers N ad λ > 0, ( λ 2 P(X N X 0 λ exp 2. N i=1 c2 i Proof of Theorem 3. We may assume without loss of geerality that δ > 0 is small eough to esure δ 1 26l(r l 2 log(1/δ ad l log(1/δ log 2 as this oly makes our task harder. Set m 0 = 26l(r l 2 δ 2 log(1/δ. Note that this implies that 2l log m 0 4l log ( 26l(r l 2 δ 2 log(1/δ 12l log(1/δ. (1 Fix m m 0. Let m m 0 ad ε = m l /2. the electroic joural of combiatorics 25(2 (2018, #P2.18 4

6 Suppose G = (V, E is a r-graph o vertices with δ ε l (G p( l. We claim that it cotais a iduced subgraph o m vertices with miimum l-degree at least (p δ ( m l. For p δ, we have othig to prove, so we may assume that 1 p > δ. Call a l-subset S V (l poor if deg(s < p ( l, ad rich otherwise. Let P be the collectio of all poor l-subsets. By our assumptio o δl ε(g, P ε( l. As each poor l-subset is cotaied i ( l m l m-subsets, it follows that there are at least ( ( l P > ( 1 εm l( = 1 ( m m l m 2 m m-subsets of vertices which do ot cotai ay poor l-subsets. Give a l-subset S V (l \ P, we call a m-subset T of V bad for S if S T ad ( N(S T ( (p δ m l. Let φs be the umber of bad m-subsets for S. We claim that ( ( l φ S exp δ2 m. (3 m l 2(r l 2 Observe that { φ S = T (V \ S (m l : ( } N(S T ( m l (p δ. r l Let X be the radom variable N(S T (, where T is a (m l-subset of V \S picked uiformly at radom. We cosider the vertex exposure martigale o T. Let Z i be the ith exposed vertex i T. Defie X i = E(X Z 1,..., Z i. Note that {X i : i = 0, 1,..., m l} is a martigale ad X 0 p ( m l Lemma 8 applied with λ = δ ( m ( ( m l P X m (p δ r l P(X m X 0 λ exp (2. Moreover, Xi X i 1 ( ( m l 1 1 < m 1 1. Thus, by ad ci = ( m 1 1, we have ( ( δ 2 m 2 ( ( δ 2 m ( exp δ2 m 2(r l 2. 2m ( m = 2(r l Hece (3 holds. A m-subset T of V is called bad if it is bad for some S V (l \ P. The umber of bad m-subsets is at most ( ( ( ( ( ( l φ S exp δ2 m m = exp δ2 m l m l 2(r l 2 m l 2(r l 2 S V (l \P ( ( ( m l 0 exp δ2 m 0 m 2(r l ( 2 exp ( l log(1/δ 1 m 2 ( m m, exp (2l log m 0 13l log(1/δ the electroic joural of combiatorics 25(2 (2018, #P2.18 5

7 where the last three iequalities hold by our choice of m 0, by iequality (1, ad by our assumptio o δ, respectively. Together with (2, this shows there exists a m-subset iside which there is o poor l-subsets ad i which every rich l-subset has degree at least (p δ ( m l. Such a set clearly gives us a iduced subgraph of G o m vertices with miimum l-degree at least (p δ ( m l. 3 Cocludig remarks A 3-graph G is (p, ε, (1, 1, 1-quasiradom if for every triple of sets of vertices X, Y ad Z V, the umber e 1,1,1 (X, Y, Z of triples (x, y, z X Y Z such that xyz E(G satisfies e 1,1,1 (X, Y, Z p X Y Z εv(g 3. Defie π (1,1,1 qr (F aalogously to π (1,2 qr (F. Note that π (1,2 qr (F π (1,1,1 qr (F π(f for all 3-graph families F. A obvious ope questio is whether we have π (1,1,1 qr (F π 2 (F. Eve more: ca oe always extract subgraphs with large miimum codegree from (1, 1, 1- quasiradom graphs? Eve obtaiig large subgraphs with o-zero miimum codegree remais a ope problem for this weaker otio of quasiradomess. Ackowledgemets The authors are grateful for a Scheme 4 grat from the Lodo Mathematical Society which allowed Victor Falgas-Ravry to visit Alla Lo i Birmigham i July 2016, whe this research was doe. Further, the authors would like to thak the aoymous referees for their careful work ad helpful suggestios, which led to cosiderable improvemets i the paper i particular, their commets led us to state ad prove a much more geeral form of Theorem 3 tha we had i the first versio of this paper. Refereces [1] N. Alo ad J.H. Specer. The Probabilistic Method. Wiley-Itersci. Ser. Discrete Math. Optim., Joh Wiley & Sos, Hoboke, NJ, [2] A. Czygriow ad B. Nagle. A ote o codegree problems for hypergraphs. Bull. Ist. Combi. Appl, 32:63 69, [3] P. Erdős ad A. Hajal. O Ramsey-like theorems: problems ad results. I Combiatorics: beig the proceedigs of the Coferece o Combiatorial Mathematics held at the Mathematical Istitute, Oxford 1972, pages Southed-o-Sea: Istitute of Mathematics ad its Applicatios, [4] V. Falgas-Ravry. O the codegree desity of complete 3-graphs ad related problems. Electro. J. Combi., 20(4:#P28, the electroic joural of combiatorics 25(2 (2018, #P2.18 6

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