Hypergrph regulrity nd qusi-rndomness Brendn Ngle Annik Poerschke Vojtěch Rödl Mthis Schcht Astrct Thomson nd Chung, Grhm, nd Wilson were the first to systemticlly study qusi-rndom grphs nd hypergrphs, nd proved tht severl properties of rndom grphs imply ech other in deterministic sense. Their concepts of qusi-rndomness mtch the notion of ε-regulrity from the erlier Szemerédi regulrity lemm. In contrst, there exists no nturl hypergrph regulrity lemm mtching the notions of qusi-rndom hypergrphs considered y those uthors. We study severl notions of qusi-rndomness for 3- uniform hypergrphs which correspond to the regulrity lemms of Frnkl nd Rödl, Gowers nd Hxell, Ngle nd Rödl. We estlish n equivlence mong the three notions of regulrity of these lemms. Since the regulrity lemm of Hxell et l. is lgorithmic, we otin lgorithmic versions of the lemms of Frnkl Rödl specil cse thereof) nd Gowers s corollries. As further corollry, we otin tht the specil cse of the Frnkl Rödl lemm which we cn mke lgorithmic) dmits corresponding counting lemm. This corollry follows y the equivlences nd tht the regulrity lemm of Gowers or tht of Hxell et l. dmits counting lemm.) 1 Introduction Thomson [18, 19] nd Chung, Grhm, nd Wilson [5] were the first to systemticlly study qusi-rndom grphs nd hypergrphs, nd proved tht severl properties of rndom grphs imply ech other in deterministic sense. Recently, nd in connection with hypergrph regulrity lemms, relted concepts of qusirndomness for hypergrphs were introduced. We focus to the 3-uniform hypergrph regulrity lemms of Frnkl nd Rödl [8], Gowers [9] nd Hxell, Ngle nd Deprtment of Mthemtics nd Sttistics, University of South Florid, Tmp, FL 3360, USA, ngle@mth.usf.edu. Reserch ws supported y NSF grnt DMS 0639839. Deprtment of Mthemtics nd Computer Science, Emory University, Atlnt, GA 303, USA, poersc@emory.edu. Deprtment of Mthemtics nd Computer Science, Emory University, Atlnt, GA 303, USA, rodl@mthcs.emory.edu. Reserch ws supported y NSF grnts DMS 030059 nd DMS 0800070. Institut für Informtik, Humoldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germny, schcht@informtik.hu-erlin.de. Rödl [11, 1]. In this pper, we discuss the reltion of these hypergrph concepts to those suggested erlier, nd we estlish n equivlence mong these properties see Corollry.1). As consequence, we infer lgorithmic versions of the regulrity lemms for 3-uniform hypergrphs of Frnkl nd Rödl nd of Gowers see Corollry.) using tht the lemm of Hxell et l. is lgorithmic). Perhps the most importnt feture of these three regulrity lemms is tht they ll dmit corresponding counting lemm which estimtes the numer of ny fixed suhypergrph in n ppropritely qusirndom environment). Strictly speking, our lgorithm nd equivlence) for Frnkl nd Rödl s lemm cn only consider specil cse of their lemm) for which no corresponding counting lemm hd een otined efore. A further corollry of our work shows tht, nonetheless, this specil cse which we cn mke lgorithmic) does dmit counting lemm see Corollry.3). 1.1 Qusi-rndom grphs. We egin our discussion with some results on qusi-rndom grphs from the ppers of Thomson [18, 19] nd Chung, Grhm nd Wilson in their influentil pper [5]. We consider the grph properties of uniform edge distriution disc), devition dev), nd C 4 -minimlity cycle). We sy sequence of grphs G n = V n, E n )) n N with V n = n nd density eg n )/ n ) = d stisfies property disc: if eu) d U dev: if u,v V n i,j {0,1} ) = on ) for every U V n, d i j d 1) i+j N i u) N j v) = on3 ), cycle: if the numer of ordered cycles of length four in G n is t most d 4 n 4 + on 4 ), where we denote y N 1 u) the neighourhood Nu) of u nd y N 0 the set V n \ Nu) of non-djcent vertices of u, nd where n ordered cycle of length 4 is sequence of distinct vertices v 1, v, v 3, v 4 ) of V n where {v i, v j } E n whenever i j = 1, 3. The three properties ove re ll equivlent [5]. Note tht when d = 1/, it follows from the definition tht dev holds if, nd only if, G n contins pproximtely) s mny sugrphs of C 4 the 4-cycle) hving oddly mny
edges s it does sugrphs of C 4 hving evenly mny edges. For densities d 1/, one scles the weights of these sugrphs ppropritely. More precisely, for grph G n = V n, E n ) of density d, we note tht dev is equivlent to 1.1) u 0,u 1 V n v 0,v 1 V n i {0,1} j {0,1} gu i, v j ) = on 4 ), where gu, v) = 1 d if {u, v} E n nd gu, v) = d if {u, v} E n. The qusi-rndom concepts ove re closely relted to the erlier notion of ε-regulrity, centrl to Szemerédi s regulrity lemm [17] see Theorem 3.). Roughly speking, the regulrity lemm sserts tht the vertex set of ny grph cn e prtitioned into ounded numer of clsses in such wy tht most of its resulting induced iprtite sugrphs stisfy iprtite version of disc see disc in Definition 1.1) nd so, y the forementioned equivlence, they lso stisfiy iprtite versions of dev nd cycle). The equivlence ove ws used in [1, ] to derive the lgorithmic version of Szemerédi s regulrity lemm. Indeed, nively checking disc requires exponentil time, while cycle or dev) cn e verified in polynomil time nd checking disc ws the centrl difficulty in mking Szemerédi s originl proof constructive). We now consider four pproches to possile generliztions of disc, dev, nd cycle to 3-uniform) hypergrphs. The first three pproches will lck importnt properties which held in the cse of grphs. In Section 1.5 we will finlly stte the pproprite generliztion nd then in Secion we stte our min results. 1. Strightforwrd generliztion. The concepts disc, dev, nd cycle hve nturl counterprts for 3-uniform hypergrphs s well s for k-uniform hypergrphs). It turned out tht finding the pproprite generliztion is not strightforwrd. For exmple, let s sy tht 3-uniform, n-vertex, hypergrph H n with d n 3) hyperedges stisfies wek-disc, if eu) d ) U 3 = on 3 ) for ll susets U V H n ), nd let s sy tht H n stisfies oct if its numer of ordered octrhedr is symptoticlly miniml d 8 n 6 +on 6 ). Here, the octhedron is the complete 3-prtite 3-uniform hypergrph K 3),, hving two vertices per clss, nd n ordering of K 3),, corresponds to leling of its vertices.) Then wek-disc nd oct re not equivlent. Indeed, let H n = K 3 Gn, 1/)) e the 3-uniform hypergrph whose triples correspond to tringles of the rndom grph Gn, 1/) on n vertices, where the edges of Gn, 1/) pper independently with proility 1/. Then, w.h.p., H n stisfies wek-disc with density d = 1/8 + o1) nd contins 1/8) 4 n 6 + on 6 ) ordered copies of K 3),,. However, ll n-vertex 3-uniform hypergrphs of density d = 1/8 contin t lest 1/8) 8 n 6 + on 6 ) ordered copies of K 3),,, nd this lower ound is relized y the rndom 3-uniform hypergrph on n-vertices whose edges re independently included with proility 1/8. Similr counterexmples exist for the devition property, which for 3-uniform hypergrph H n = V n, E n ) of density d is defined s Y 1.) dev : hu i, v j, w k ) = on 6 ), u 0,u 1 v 0,v 1 w 0,w 1 i,j,k {0,1} where hu, v, w) = 1 d if {u, v, w} E n nd hu, v, w) = d if {u, v, w} E n. We mention tht one cn prove hypergrph regulrity lemm whose regulrity concept corresponds to wek-disc. An unstisfying feture of such lemm is tht it cn t, in principle, dmit corresponding counting lemm. There re no known hypergrph regulrity lemms corresponding to oct or dev, s we ve defined them ove. 1.3 A refined pproch to disc. Frnkl nd Rödl suggested the following concept of uniform edge distriution see lso [3, 4]). Sy tht n n-vertex 3-uniform hypergrph H n = V n, E n ) of density d stisfies disc if E n K 3 G) d K 3 G) = on 3 ) holds for ll grphs G with vertex set V n, where K 3 G) denotes the collection of triples of vertices of V n which spn tringle K 3 in G. For d = 1/, it ws shown in [4] tht disc just defined) nd dev nd oct defined ove) re ll equivlent see lso [13] for d 1/). In the definition ove, we my view the hypergrph H n = V n, E n ) s suset of the tringles of the complete grph K n. Similrly to how Szemerédi s regulrity lemm prtitions the vertex set of grph, the recent regulrity lemms for 3-uniform hypergrphs lso prtition the set of pirs of vertices. As consequence, it is necessry to consider notions of qusi-rndomness which involve not only the hypergrph H n = V n, E n ), ut lso n underlying grph G for which E n K 3 G). 1.4 Asolute qusi-rndom properties. The discussion ove leds to the following concepts, which were prtly studied in [13]. To egin our presenttion, we stte the iprtite versions of disc, dev, nd cycle for grphs. Definition 1.1. Let ε > 0 nd let G = U V, E) e iprtite grph with U = V = n nd density eg)/n = d ± ε. We sy G hs the property disc ε): if e G U, V ) d U V εn for ll U U nd V V ;
dev ε): if u 0,u 1 U v 0,v 1 V i {0,1} j {0,1} gu i, v j ) εn 4, where gu, v) = 1 d if {u, v} E nd gu, v) = d if {u, v} E; cycle ε): if G contins t most d 4 n ) +εn 4 4-cycles. We now define corresponding notions for 3-uniform hypergrphs H with underlying 3-prtite grphs G. Definition 1.. Let ε > 0 nd let G = G 1 G 13 G 3 e 3-prtite grph with 3-prtition V G) = U V W, U = V = W = n, nd let H e 3-uniform hypergrph where EH) K 3 G). Let G ij e of density d ± ε for 1 i < j 3 nd let eh) = d 3 K 3 G), i.e., H hs reltive density d 3 w.r.t. G. We sy H, G) hs the property disc 3 ε): if G ij hs disc ε) for 1 i < j 3 nd EH) K 3 G ) d 3 K 3 G ) εn 3 for ll sugrphs G of G; dev 3 ε): if G ij hs dev ε) for 1 i < j 3 nd Y h H,Gu i, v j, w k ) εn 6, u 0,u 1 U v 0,v 1 V w 0,w 1 W i,j,k {0,1} where 8 >< 1 d 3, if {u, v, w} EH) h H,Gu, v, w) = d 3, if {u, v, w} K 3G) \ EH) >: 0, otherwise; oct 3 ε): if G ij hs cycle ε) for 1 i < j 3 nd H contins t most d 8 3d 1 n 3 ) + εn 6 copies of K 3),,. We refer to pirs H, G) stisfying the properties in Definition 1. with ε d, d 3 s solute qusi-rndom, since the mesure of qusi-rndomness ε of the hypergrph H is smller thn the solute) density of H, which is essentilly d 3 d 3. It ws shown in [13] see lso [15, Theorem.]) tht for every d 3, d, nd ε > 0 there exists δ > 0 such tht if pir H, G) stisfies disc 3 δ), then it lso stisfies oct 3 ε). In other words, disc 3 implies oct 3, nd the rguments from [4] nd [13] cn e extended to show tht indeed ll three notions disc 3, dev 3, nd oct 3 re equivlent in this sense. Note tht the properties in Definition 1. ecome meningless if ε min{d, d 3 }, since then the error term is lrger thn the min term. However, in ll known regulrity lemms, the condition tht ε < min{d, d 3 } in fct ε min{d, d 3 }) cnnot e gurnteed. More precisely, the mesure of qusi-rndomness ε of the 3- uniform hypergrph will typiclly e lrger thn the density d of the uxillry underlying grphs in the regulr prtition of those lemms. We therefore need refinement of the properties from Definition 1., which leds to the following reltive concepts of qusirndomness. For regulr prtition whose typicl locks disply ε min{d, d 3 }, one must pertur the edge set of the input hypergrph, which will e discussed in Theorem 3.3 elow cf. [7, 16]).) 1.5 Reltive qusi-rndom hypergrphs. The recent regulrity lemms for 3-uniform hypergrphs of Frnkl Rödl [8], Gowers [9], nd Hxell et l. [11, 1] re sed on the following notions of qusi-rndomness, in which the qusi-rndomness of H nd G re mesured y ε 3 nd ε, resp., nd where it will typiclly e the cse tht d 3 ε 3 d ε. Definition 1.3. Let ε 3, ε > 0 nd G = G 1 G 13 G 3 e 3-prtite grph with 3-prtition V G) = U V W, U = V = W = n, nd let H e 3-uniform hypergrph with EH) K 3 G). Let G ij e of density d ± ε for 1 i < j 3 nd let eh) = d 3 K 3 G). We sy H, G) hs the property disc 3 ε 3, ε ): if G ij hs disc ε ) for 1 i < j 3 nd EH) K 3 G ) d 3 K 3 G ) ε 3 d 3 n 3 for ll G G; dev 3 ε 3, ε ): G ij hs dev ε ) for 1 i < j 3 nd for the function h H,G u, v, w), defined s in Definition 1., we hve Y u 0,u 1 U v 0,v 1 V w 0,w 1 W i,j,k {0,1} h H,Gu i, v j, w k ) ε 3d 1 n 6 ; oct 3 ε 3, ε ): if G ij hs cycle ε ) for 1 i < j 3 nd H contins t most d 8 3d 1 n ) 3 + ε3 d 1 n 6 copies of K 3),,. We refer to pirs H, G) stisfying the properties in Definition 1.3 with ε d ε 3 d 3 s reltive qusirndom since here the mesure of qusi-rndomness ε 3 of the hypergrph H is only smller thn the reltive density d 3 of H w.r.t. G. 1.6 Hypergrph regulrity lemms. We stte the regulrity lemm for 3-uniform hypergrphs of Gowers [9]. The centrl concept of qusi-rndomness in this lemm is dev 3. Theorem 1.1. For every ε 3 > 0, every function ε : N 0, 1], nd every t 0 N, there exist positive integers T 0 nd n 0 so tht for every 3-uniform hypergrph
H = V, E) on n n 0 vertices, there exist vertex prtition V = V 1... V t, where V 1 V t V 1 +1 nd t 0 t T 0, nd prtition of pirs of the complete iprtite grphs K[V i, V j ], 1 i < j t, given y K[V i, V j ] = G ij 1... G ij l, where l T 0, so tht the following holds. All ut ε 3 n 3 triples {x, y, z} V 3) stisfy tht whenever {x, y, z} K 3 G ij jk G G ik c ) = K 3 G ), for some 1 i < j < k t nd,, c) [l] 3, then H, G ) stisfies dev 3ε 3, ε l)) with reltive density H / K 3G ) of H with respect to G nd the densities of G ij, G jk, nd Gik c eing 1/l, where H hs edge set EH) K 3G ). If we replce dev 3 in Theorem 1.1 y disc 3 or oct 3, then we resp.) otin the hypergrph regulrity lemms of Frnkl nd Rödl [8] nd of Hxell et l. [11, 1]. Remrk 1.1. Theorem 1.1 differs slightly from the version proved y Gowers [9] in tht the originl does not require most iprtite grphs G ij to hve density close to 1/l. The dditionl ssertion we hve stted cn e otined long similr lines to [8]. We point out tht the regulrity lemm of Frnkl nd Rödl is stronger thn we hve quoted ove. It sserts the existence of prtition such tht most H, G ) stisfy the following stronger vrint disc 3,r of disc 3 where r cn depend on l nd t). For H nd G s in Definition 1.3 nd n integer r 1, we sy H, G) stisfies disc 3,r ε 3, ε ) if i ) G ij hs disc ε ) for 1 i < j 3 nd ii ) EH) S i [r] K3Gi) d3 S i [r] K3Gi) ε3d3 n 3 for ll fmilies of sugrphs G 1,..., G r of G. Clerly, disc 3,1 = disc 3, ut otherwise disc 3,r is stronger thn disc 3. Dementiev, Hxell, Ngle nd Rödl [6, Theorem 3.5] proved tht oct 3 disc 3,r when r is lrge. New results The min new result is the equivlence of the notions of qusi-rndom hypergrphs from Definition 1.3. Theorem.1. For ll d 3, ε 3 > 0, there exists δ 3 > 0 such tht for ll d, ε > 0, there exist δ > 0 nd n 0 such tht the following holds. Let G = G 1 G 13 G 3 e 3-prtite grph with 3- prtition V G) = U V W, U = V = W = n n 0, nd let H e 3-uniform hypergrph where EH) K 3 G). Let G ij e of density d ± δ, 1 i < j 3, nd let eh) = d 3 K 3 G). i ) If H, G) stisfies disc 3 δ 3, δ ), then it lso stisfies oct 3 ε 3, ε ), i.e., disc 3 oct 3. ii ) If H, G) stisfies oct 3 δ 3, δ ), then it lso stisfies dev 3 ε 3, ε ), i.e., oct 3 dev 3. We prove the ssertions i ) nd ii ) of Theorem.1 in Sections 3 nd 4, resp. We continue with few immedite corollries of our min result. First, the ssertion of i ) ove directly confirms Conjecture 3.8 of Dementiev et l. [6]. They proved [6, Theorem 3.6] oct 3 disc 3, in which cse the ssertion of i ) ove gives oct 3 disc 3. However, direct consequence of the counting lemm of Gowers [9, Theorem 6.8] more precisely, [10, Corollry 5.3]) gives dev 3 oct 3. As such, we hve the following corollry. Corollry.1. The properties disc 3, dev 3, nd oct 3 re equivlent. Reclling from Dementiev et l. [6] tht oct 3 disc 3,r when r is lrge), Corollry.1 llows us to extend their work to sy tht dev 3 disc 3,r. From the lgorithmic regulrity lemm of Hxell et l. [11, 1] sed on oct 3 ), the equivlence ove implies lgorithmic versions of the 3-uniform hypergrph regulrity lemms of Gowers [9] nd Frnkl Rödl [8] when r = 1). Corollry.. There exists n lgorithm with running time On 6 ), which constructs the prtitions of vertices nd pirs from Theorem 1.1. Strictly speking, n lgorithmic version for r = 1 of the Frnkl Rödl regulrity lemm ws lredy stted y Dementiev et l. in [6, Theorem 3.10]. However, t the time of tht nnouncement, no corresponding counting lemm ws known. By ppeling to the counting lemm of Gowers [9] or Hxell et l. [11, 1], the equivlence ove implies counting lemm pplicle to the specil cse r = 1. Corollry.3. For every p N nd ξ, d 3 > 0 there exists δ 3 > 0 such tht for every d > 0 there exist δ > 0 nd n 0 such tht the following holds. Let G = 1 i<j p Gij e p-prtite grph with vertex prtition V 1... V p where V 1 = = V p = n n 0 nd let H e 3-uniform hypergrph with EH) K 3 G). Let G ij e of density d ± δ, 1 i < j p nd let eh ) = d 3 K 3 G ) for ll 1 i < j < k p, where G = G[V i, V j, V k ] nd H = H K 3 G ). Suppose, moreover, tht ech H stisfies disc 3 δ 3, δ ), 1 i < j < k p. Then the numer K p H) of complete, 3-uniform hypergrphs on p vertices in H stisfies K p H) = 1 ± ξ)d p 3) 3 d p ) n p.
3 Uniform edge distriution implies minimlity In this section, we prove prt i ) of Theorem.1. The proof is sed on the sme impliction in the solute setting, where roughly speking we will trnsfer the known impliction disc 3 oct 3 from the solute setting to the reltive setting. Similr ides were used in [14].) For tht we will use Szemerédi s regulrity lemm for grphs see Theorem 3.) nd the regulr pproximtion lemm for 3-uniform hypergrphs see Theorem 3.3). We stte these uxilry results in the next section nd prove prt i ) of Theorem.1 in Section 3.. 3.1 Auxiliry results. We will use the following proposition, which follows from [13, Theorem 6.5] see lso [15, Theorem.]). Theorem 3.1. For ll d 3, ε > 0, there exist δ > 0 nd n 0 such tht the following holds. Let D e 3-prtite, 3-uniform, hypergrph on the vertex prtition U V W, U = V = W = n n 0, nd let ed) = d 3 ±δ)n 3. If D, K[U, V, W ]) stisfies disc 3 δ), then D, K[U, V, W ]) hs oct 3 ε), where K[U, V, W ] denotes the complete triprtite grph on U V W. Note tht Theorem 3.1 drws the sme conclusion s i ) of Theorem.1, ut in the solute setting. For the trnsfer of this result to the reltive setting, we will employ the regulr pproximtion lemm for 3-uniform hypergrphs from [16], Theorem 3.3, nd Szemerédi s regulrity lemm for grphs [17], Theorem 3., which we stte elow ut in opposite order). Theorem 3.. For ll µ > 0 nd integers t nd M, there exist S 0 nd n 0 such tht for every fmily of grphs F 1,..., F M on the sme vertex set V with V = n n 0 nd n eing multiple of S 0!) nd for ny given prtition V = V 1... V t, V i = n/t for i [t], there exists refinement V = i [t],j [s] V i,j, with V i,j = n/ts) nd s S 0, such tht for ll ut µt s pirs {{i, j}, {k, l}}, 1 i < j t, 1 k, l s, the induced iprtite grphs F m [V i,j, V k,l ] stisfy disc µ) for ll m = 1,..., M. Next we stte the regulr pproximtion lemm for 3-uniform hypergrphs see [16, Lemm 4.] or [14, Theorem 54]). Roughly speking, it sserts tht for every 3-uniform hypergrph H, there exists hypergrph H otined from H y dding or deleting few hyperedges from H, so tht H dmits vertex prtition nd prtition of pirs, s in Theorem 1.1, with the stronger property tht for ll locks of the prtition, the hypergrph H stisfies the solute disc 3 property from Definition 1.. Theorem 3.3. For ll d, ν > 0 nd every function ϱ: N 0, 1], there exist ε 0 > 0 nd T 0 so tht the following holds. Let G = G 1 G 13 G 3 e 3-prtite grph with 3-prtition V G) = U V W, U = V = W = n n 0 where n is multiple of T 0!) nd let H e 3-uniform hypergrph with EH) K 3 G). Let G ij stisfy disc ε 0 ) with density d for 1 i < j 3. Then there exist integers t nd l T 0 nd ) vertex prtition U = i [t] U i, V = j [t] V j, nd W = k [t] U k, with U i = V j = W k = n/t for i, j, k [t], ) prtition of pirs of the induced iprtite grphs G 1 [U i, V j ], G 13 [U i, W k ], nd G 3 [V j, W k ], i, j, k [t], given y G 1 [U i, V j ] = P Ui,Vj 1... P Ui,Vj l, G 13 [U i, W k ] = P Ui,W k 1... P Ui,W k l, nd G 3 [V j, W k ] = P Vj,W k 1... P Vj,W k, nd c ) 3-prtite, 3-uniform hypergrph H on the sme vertex set U V W such tht the following holds: I) EH) E H) νn 3 nd II) for ll 1 i < j < k t nd,, c) [l] 3 the pir H, P ) hs disc 3ϱt, l)) with reltive density E H ) / K 3P ) nd the densities of the involved iprtite grphs eing d /l, where P = P Ui,Vj P Ui,W k P Vj,W k c nd H = H K 3 P ). The min difference etween Theorems 1.1 nd 3.3 concerns the degree of qusi-rndomness of H, P ) in Theorem 3.3) nd H, G ) in Theorem 1.1). Theorem 3.3 gurntees tht, t the cost of ltering only few triples glolly), the mesure ϱt, l) of qusirndomness cn e much smller thn 1/tl), while Theorem 1.1 cn only gurntee the mesure ε 3 of qusirndomness s fixed constnt where t nd l depend of ε 3 ). On the other hnd, in Theorem 1.1, the qusirndom property holds directly for H, while in Theorem 3.3, it only pplies to the chnged hypergrph H. 3. Proof of i ) of Theorem.1. We now prove ssertion i ) of Theorem.1. Proof. disc 3 oct 3 ) Let d 3, ε 3 > 0 e given nd let δ e the constnt ensured y Theorem 3.1 for d 3 nd ε = ε 3 /4. Without loss of generlity, we my ssume tht δ ε d 8 3/8. For Theorem.1, we set δ 3 = δ /4 nd let δ 0 δ 3. Then, for given d nd ε > 0, we set 0 < ν min{δ 3 d 3 d 3, ε 3 d 8 3d 1 /4, δ d 3 /} l
nd ) 3 δ 0 0 < ϱt, l) S 0 µ δ 0 /l, M = 3t, l, t) i.e., ϱt, l) tends fster to 0 when t nd l tend to infinity) thn δ 0 /S 0 ) 3, where S 0 t, l) is given y Szemerédi s regulrity lemm, Theorem 3., pplied with 0 < µ δ 0 /l, M = 3t l, nd t. Finlly, let 0 < δ ε 0 min ϱt, l), t [T 0],l [T 0] where ε 0 nd T 0 re given y the regulr pproximtion lemm, Theorem 3.3, pplied with ν nd ϱ, ). Moreover, we choose δ smll enough so tht disc δ ) cycle ε ) for iprtite grphs of density d. For these constnts nd sufficiently lrge n let H, G) e pir stisfying disc 3 δ 3, δ ) s given in Theorem.1. We hve to show tht H, G) stisfies oct 3 ε 3, ε ). We first pply Theorem 3.3, with ν nd ϱt, l) ove, to H nd G nd otin integers t nd l T 0, vertex prtition, prtition of pirs, nd hypergrph H s stted in ) c ) in Theorem 3.3 with properties I) nd II). We wnt to pply Theorem 3.1. For this we construct dense 3-prtite, 3-uniform, hypergrph D on the sme vertex set U V W, which we view s suhypergrph of K 3 K[U, V, W ]) the tringles of K[U, V, W ]. Roughly speking, we will construct D y mimicking the prtition of vertices nd pirs of H, which we otined from Theorem 3.3. For tht we will consider the sme vertex prtition, ut replce every grph P Ui,Vj of density d /l similrly, P Ui,W k nd P Vj,W k ) y rndom grph B Ui,Vj of density 1/l nd for every B we let the edges of D e rndom suset of K 3 B ) with reltive density mtching the one of H w.r.t. P. As consequence of this construction the hypergrph D will hve solute density d 3 ± ν note H only hs reltive density d 3 w.r.t. G) nd we will show tht D, K[U, V, W ]) stisfies disc 3 δ ) see Clim 1). Hence, Theorem 3.1 implies tht D, K[U, V, W ]) will lso stisfy oct 3 ε 3 /4), which estimtes the numer of octhedr in D. On the other hnd, we will show tht the construction of D yields #{K 3),, D} d1 #{K 3),, H} see Clim ). From tht we will infer tht H, G) stisfies oct 3 ε 3, ε ), since EH) E H) νn 3 ε 3 d 8 3d 1 n 3 /4. We now give the detils of this pln. For the construction of D, we will mimic the prtition of vertices nd pirs which we otined for H fter we pplied Theorem 3.3. Recll we tke the vertex set of D the sme s of H, i.e., U V W, where there exists prtition of U = U 1... U t, V = V 1... V t, nd W = W 1... W t. Now for ll i, j [t], consider rndom prtition of the edge set of K[U i, V j ] into l prts K[U i, V j ] = B Ui,Vj 1... B Ui,Vj l. Define the grphs B Ui,W k nd B Vj,W k c for i, j, k [t] nd, c [l] nlogously. We my think of the grph B Ui,Vj s plying similr role for D s P Ui,Vj does for H. Note, however, tht the density of B Ui,Vj is 1/l, while the density of P Ui,Vj is d /l. To define the edges of D, fix i, j, k [t] nd,, c [l] nd set B = BUi,Vj B Ui,W k B Vj,W k c. Let D, the suhypergrph of D induced on K 3 B ), e rndom suset of K 3 B ), where ech triple {u, v, w} K 3 B ) is chosen to e n edge in D independently with proility d H P ) = E H ) / K 3P ). In other words, we construct D in such wy tht the reltive density of D w.r.t. B, i.e., dd B ), is very close to d H P ), i.e., the reltive density of H w.r.t.. We will verify two clims, Clim 1 nd, for D. P Clim 1. D, K[U, V, W ]) stisfies disc 3 δ ) nd ed) = d 3 ± δ /)n 3 with proility 1 o1). Proof. Consider n ritrry sugrph F of K[U, V, W ], which we view s the union of 3t l grphs of the form F Ui,W k F Ui,Vj = F B Ui,Vj, = F B Ui,W k, nd F Vj,W k c = F B Vj,W k c. We pply Szemeredi s regulrity lemm, Theorem 3., to ll such 3t l grphs. This wy we otin refinement of the vertex prtition on U V W, nd ech F Ui,Vj is split into s typiclly) qusi-rndom iprtite grphs. For ech of these 3t ls grphs, sy F = F Ui,Vj [U i,p, V j,q ] B Ui,Vj [U i,p, V j,q ] = B with p, q [s], we consider rndom sugrph Q P = P Ui,Vj [U i,p, V j,q ], where we include every edge of P independently with proility ef )/eb ), i.e., Q hs pproximtely the sme reltive density compred to P, s the grph F hs w.r.t. B. Finlly, we consider the union of ll such Q. So let Q = i,j [t] p,q [s] [l] Q Q Ui,p,W k,r i,k [t] p,r [s] [l] j,k [t] q,r [s] c [l] Q Vj,q,W k,r c
e the union of ll these rndom grphs. We will show tht w.h.p. 3.3) K3 Q) d 3 K 3 F ) δ 8 d3 n 3 nd 3.4) E H) K 3 Q) d 3 ED) K 3 F ) δ From 3.3) nd 3.4) we infer 8 d3 n 3 ED) K 3 F ) d 3 K 3 F ) E H) K 3 Q) d 3 d 3 K 3 Q) d 3 + δ 4 n3 EH) K 3 Q) d 3 d 3 K 3 Q) d 3 + δ 4 n3 + ν d 3 δ 4 n3 + δ 4 n3 + ν d 3 n 3 δ n 3, since H, G) stisfies disc 3 δ 3, δ ) with δ 3 δ /4 nd since EH) E H) νn 3 δ d 3 n 3 /. Since F ws n ritrry sugrph of K[U, V, W ], this implies tht D, K[U, V, W ]) stisfies disc 3 δ ). For the proof of 3.3) we consider triprtite grphs nd F,pqr Q,pqr = F = Q F Ui,p,W k,r Q Ui,p,W k,r F Vj,q,W k,r c Q Vj,q,W k,r c. Suppose the iprtite sugrphs of F,pqr stisfy disc µl)) ll ut µt s do) nd hve density δ 0 /l. Then we cn ppel to the counting lemm for grph tringles nd infer tht the numer of tringles in stisfies F,pqr 1 ± ξ µ ) ef Ui,p,V j,q ) ef Ui,p,W k,r ) ef Vj,q,W k,r c ) n/st)) 3, where ξ µ 0 s µ 0. On the other hnd, since P Ui,Vj stisfies disc ϱt, l)), we hve tht P stisfies disc s ϱt, l)) with density d /l ± s ϱt, l) + δ ). Consequently, since Q is rndom sugrph it stisfies disc s ϱt, l) + o1)) s long s the density of F is 1/ log n). Moreover, if the density of is t lest δ 0 /l, we hve tht F eq ) = d ± s ϱt, l) + δ + o1)))ef ). Consequently, if the iprtite sugrphs of F,pqr hve density δ 0 /l, then we hve, gin due to the tringle counting lemm, K 3 Q,pqr ) = 1 ± ζ s ϱ + δ ))d 3 ef Ui,p,V j,q )ef Ui,p,W k,r )ef Vj,q,W k,r c ) n/st)) 3, n 3 where ζ s ϱ 0 s sϱ 0. In other words, we hve shown tht if the iprtite sugrphs of F,pqr stisfy disc µl)) nd hve density δ 0 /l, then K 3 Q,pqr ) = 1 ± ζ s ϱ + ξ µ + δ + o1)))d 3 K 3 F,pqr ). Finlly, the first ssertion of 3.3) follows from the choice of δ 0, µl) δ 0 /l, ϱt, l) 1/S 0, nd the fct tht ll ut µt s iprtite grphs F stisfy disc µl)). Noting tht, if the iprtite sugrphs of F,pqr stisfy disc µl)) nd hve density δ 0 /l, then,pqr H, P,pqr ) stisfies disc 3 s 3 ϱt, l)/δ0) 3 nd ppeling to the rndom construction of D, we infer tht,pqr d H Q ) = dd F,pqr ) ± s 3 ϱt, l)/δ0 3 + o1) nd the second ssertion of 3.3) follows from the discussion ove. Clim. With proility 1 o1) we hve #{K 3),, H} 1 + o1))d 1 #{K 3),, D}. Proof. Apply the counting lemm from [13, Theorem 6.5] to H to count the numer of octhedr. More precisely, pply the dense counting lemm to H induced on every selection of six vertex clsses U i1, U i, V j1, V j, W k1, W k nd 1 grphs P Ui 1,Vj 1 1,..., P Ui,Vj 4,..., P Vj,W k c 4. There re t 6 l 1 such choices, nd for ech such choice, we get n estimte on the numer of octhedr of H induced on tht choice. Moreover, for ech such choice, we will consider the corresponding such selection with the iprtite grphs P,Y replced y the corresponding grph B,Y. For such selection of B-grphs, we cn estimte the numer of octhedr in D induced on those B-grphs due to the rndomness in the construction of D). The numer of octhedr in H nd D for corresponding choice of B- nd P -grphs will e equl up to fctor of d 1. Repeting this nlysis for ll pproprite t 6 l 1 choices then yields the clim. Finlly, we deduce oct 3 ε 3, ε ) for H, G) from the clims ove. Becuse of Clim 1 nd Theorem 3.1, we hve tht, w.h.p., H, G) stisfies oct 3 ε ), i.e., the numer of of octhedr in D is t most d 3 + δ ) 8 1 1 ) n 3 + ε n 6 = d 3 + δ ) 8 n 3 ) + ε n 6. Hence, we infer from the choice of δ ε d 8 3/8 nd Clim tht H, G) stisfies oct 3 ε + o1), ε ), in prticulr, H contins t most d 8 3d 1 n ) 3+ε +o1))n 6 octhedr. Note tht G ij stisfies cycle ε ) due to the choice of δ. Now it follows tht H, G) stisfies oct 3 ε 3, ε ), since ε ε 3 /4 nd since EH) E H) νn 3 ε 3 d 8 3d 1 n 3 /4, which yields tht H contins t most ε 3 d 8 3d 1 n 3 /4 n 3 octhedr more thn H.
4 Minimlity implies smll devition In this section, we prove ssertion ii ) of Theorem.1. The proof is sed on the counting lemm from Hxell et l. [1] nd on the equivlence of disc 3 nd oct 3 which ws estlished in Section 3 using the result from Dementiev et l. [6, Theorem 3.6]). More precisely, we first use these tools to derive the following induced counting lemm for suhypergrphs of the octhedron. For suocthedron O K 3),, with vertex clsses {x 0, x 1 }, {y 0, y 1 }, nd {z 0, z 1 } nd hypergrph H nd grph G with EH) K 3 G) we sy copy of O on vertex pirs {u 0, u 1 }, {v 0, v 1 }, nd {w 0, w 1 } is induced in H w.r.t. G), if {u i, v j, w k } K 3 G) for ll i, j, k = 0, 1 nd {u i, v j, w k } EH) if nd only if {x i, y j, z k } EO). Proposition 4.1. For ll ξ, d 3 > 0, there exists δ 3 > 0 such tht for ll d > 0 there exist δ > 0 nd n 0 such tht the following holds. Let G = G 1 G 13 G 3 e 3-prtite grph with 3- prtition V G) = U V W, U = V = W = n n 0 nd let H e 3-uniform hypergrph with EH) K 3 G). Let G ij e of density d ± δ for 1 i < j 3 nd let eh) = d 3 K 3 G). If H, G) stisfies oct 3 δ 3, δ ), then for every suocthedron O K 3),,, the numer of prtite) leled, induced copies of O in H w.r.t. G stisfies #{O H induced w.r.t. G} = 1 ± ξ)d eo) 3 1 d 3 ) 8 eo) d 1 n 6. Before we prove Proposition 4.1, we derive prt ii ) of Theorem.1 from it. Proof. oct 3 dev 3 ) Let d 3, ε 3 > 0 e given. We choose δ 3 > 0 smll enough so tht Propositition 4.1 holds for ξ ε 3 d 3 1 d 3 )/) 8 /. Then for given d nd ε > 0, we let δ > 0 e smll enough for Propositition 4.1 nd so tht every iprtite grph of density d with cycle δ ) lso stisfies dev ε ). Finlly, let n 0 e lrge enough so tht Propositition 4.1 nd cycle δ ) dev ε ) hold. For given pir H, G) stisfying oct 3 δ 3, δ ), we pply Propositition 4.1 for every spnning) suocthedron O K 3),,, nd since u 0,u 1 U v 0,v 1 V w 0,w 1 W i,j,k {0,1} = On 5 ) + O K 3),, h H,G u i, v j, w k ) d 3 ) 8 eo) 1 d 3 ) eo) #{O H induced w.r.t. G}, we otin u 0,u 1 U v 0,v 1 V w 0,w 1 W i,j,k {0,1} h H,G u i, v j, w k ) = On 5 )+d 8 31 d 3 ) 8 d 1 n 6 1) 8 eo) ±ξ) O On 5 )+ ε 3 d1 n 6, where we used 1) 8 eo) = 0. O K 3),, Therefore, the pir H, G) stisfies dev 3 ε 3, ε ) if n is sufficiently lrge. It is left to prove Proposition 4.1. Proof. We use the equivlence of disc 3 nd oct 3 in the following wy. Suppose H, G) stisfies disc 3 ε 3, ε ) for some densities d 3 nd d. Then it follows directly from the definition of disc 3 tht for the complement of H w.r.t. G, i.e., H = V H), K 3 G) \ EH)), H, G) stisfies disc 3 ε 3, ε ) for densities d 3 = 1 d 3 nd d. Hence, we infer from the equivlence of disc 3 nd oct 3 tht if H, G) stisfies oct 3 δ 3, δ ), then H, G) stisfies oct 3 δ 3, δ ) for some δ 3δ 3 ) 0 s δ 3 0. For the proof of Proposition 4.1 we my choose the constnts so tht min{ξ, d 3, 1 d 3 } ξ δ 3 δ 3 d δ. By the discussion ove, we my ssume tht for the given pir H, G) with oct 3 δ 3, δ ), we hve tht H, G) stisfies oct 3 δ 3, δ ). For given suocthedron O K ),,, we doule H, G) ccording to O. More precisely, let the three vertex clsses of O e {x 0, x 1 }, {y 0, y 1 }, nd {z 0, z 1 } nd let U, V, W e the vertex clsses of H nd G. First we construct new 6-prtite grph G with vertex clsses U i = U {i}, V j = V {j}, nd W k = W {k} with i, j, k = 0, 1, i.e., we tke two copies of every originl vertex clss. Moreover, let {u, i), v, j)} e n edge in G if, nd only if, {u, v} EG) similrly for {u, i), w, k)} nd {v, j), w, k)}). In other words, we otin G from G y cloning every vertex nd replcing every edge y C 4 on the corresponding cloned vertices. Note tht the construction of G is independent of O. Next we define the edges of H s follows: for u U, v V, w W, nd i, j, k = 0, 1, let {u, i), v, j), w, k)} EH ) EH), {x i, y j, z k } EO), {u, v, w} K 3G) \ EH), {x i, y j, z k } EO). In other words, H, G ) ws constructed so tht H [U i, V j, W k ], G [U i, V j, W k ]) is copy of H, G) if {x i, y j, z k } EO) nd copy of H, G) otherwise.
In ny cse, from the discussion ove, we know tht H [U i, V j, W k ], G [U i, V j, W k ]) stisfies oct 3 δ 3, δ ). Hence, the counting lemm from [1] implies tht the numer of crossing copies of K 3),, in H stisfies 1 ± ξ )d eo) 3 1 d 3 ) 8 eo) d 1 n 6. Noting, tht, due to the construction of H, this equls the numer of prtite) leled, induced copies of O in H w.r.t. G minus n error of On 5 ) for copies in H which use two copies of the sme vertex, e.g., u, 1) nd u, )), we conclude the proposition. 5 Concluding remrks The min result sserts tht for 3-uniform hypergrphs the properties disc 3, dev 3, nd oct 3 re equivlent. We elieve the sme result holds for k-uniform hypergrphs. Such equivlences would e useful to otin lgorithmic regulrity lemms for k-uniform hypergrphs. We elieve those results hold, which is work in progress. References [1] N. Alon, R. A. Duke, H. Lefmnn, V. Rödl, nd R. Yuster, The lgorithmic spects of the regulrity lemm extended strct), 33rd Annul Symposium on Foundtions of Computer Science Pittsurgh, Pennsylvni), IEEE Comput. Soc. Press, 199, pp. 473 481. 1.1 [], The lgorithmic spects of the regulrity lemm, J. Algorithms 16 1994), no. 1, 80 109. 1.1 [3] F. R. K. Chung, Qusi-rndom clsses of hypergrphs, Rndom Structures Algorithms 1 1990), no. 4, 363 38. 1.3 [4] F. R. K. Chung nd R. L. Grhm, Qusi-rndom hypergrphs, Rndom Structures Algorithms 1 1990), no. 1, 105 14. 1.3, 1.4 [5] F. R. K. Chung, R. L. Grhm, nd R. M. Wilson, Qusi-rndom grphs, Comintoric 9 1989), no. 4, 345 36. 1, 1.1 [6] Y. Dementiev, P. E. Hxell, B. Ngle, nd V. Rödl, On chrcterizing hypergrph regulrity, Rndom Structures Algorithms 1 00), no. 3-4, 93 335, Rndom structures nd lgorithms Poznn, 001). 1.6,,,, 4 [7] G. Elek nd B. Szegedy, Limits of hypergrphs, removl nd regulrity lemms. A non-stndrd pproch, sumitted. 1.4 [8] P. Frnkl nd V. Rödl, Extreml prolems on set systems, Rndom Structures Algorithms 0 00), no., 131 164. 1, 1.5, 1.6, 1.1, [9] W. T. Gowers, Qusirndomness, counting nd regulrity for 3-uniform hypergrphs, Comin. Pro. Comput. 15 006), no. 1, 143 184. 1, 1.5, 1.6, 1.1,,, [10], Hypergrph regulrity nd the multidimensionl Szemerédi theorem, Ann. of Mth. ) 166 007), no. 3, 897 946. [11] P. E. Hxell, B. Ngle, nd V. Rödl, An lgorithmic version of the hypergrph regulrity method, 46th Annul IEEE Symposium on Foundtions of Computer Science FOCS 005), 3-5 Octoer 005, Pittsurgh, PA, USA, Proceedings, IEEE Computer Society, 005, pp. 439 448. 1, 1.5, 1.6,, [1], An lgorithmic version of the hypergrph regulrity method, SIAM J. Comput. 37 008), no. 6, 178 1776. 1, 1.5, 1.6,,, 4, 4 [13] Y. Kohykw, V. Rödl, nd J. Skokn, Hypergrphs, qusi-rndomness, nd conditions for regulrity, J. Comin. Theory Ser. A 97 00), no., 307 35. 1.3, 1.4, 1.4, 3.1, 3. [14] B. Ngle, V. Rödl, nd M. Schcht, The counting lemm for regulr k-uniform hypergrphs, Rndom Structures Algorithms 8 006), no., 113 179. 3, 3.1 [15] V. Rödl nd M. Schcht, Regulr prtitions of hypergrphs: Counting lemms, Comin. Pro. Comput. 16 007), no. 6, 887 901. 1.4, 3.1 [16], Regulr prtitions of hypergrphs: Regulrity lemms, Comin. Pro. Comput. 16 007), no. 6, 833 885. 1.4, 3.1, 3.1 [17] E. Szemerédi, Regulr prtitions of grphs, Prolèmes comintoires et théorie des grphes Colloq. Internt. CNRS, Univ. Orsy, Orsy, 1976), Colloq. Internt. CNRS, vol. 60, CNRS, Pris, 1978, pp. 399 401. 1.1, 3.1 [18] A. Thomson, Pseudorndom grphs, Rndom grphs 85 Poznń, 1985), North-Hollnd, Amsterdm, 1987, pp. 307 331. 1, 1.1 [19], Rndom grphs, strongly regulr grphs nd pseudorndom grphs, Surveys in comintorics 1987 New Cross, 1987), London Mth. Soc. Lecture Note Ser., vol. 13, Cmridge Univ. Press, Cmridge, 1987, pp. 173 195. 1, 1.1