Note on the 3-graph counting lemma

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1 Note on the -grph counting lemm Brendn Ngle,,, Vojtěch Rödl, nd Mthis Schcht c, Deprtment of Mthemtics nd Sttistics, Uniersity of Ned, Reno, Reno, NV, 89557, USA Deprtment of Mthemtics nd Computer Science, Emory Uniersity, Atlnt, GA, 00, USA c Institut für Informtik, Humoldt-Uniersität zu Berlin, Unter den Linden 6, D-0099, Berlin, Germny Astrct Szemerédi s regulrity lemm proed to e powerful tool in extreml grph theory. Mny of its pplictions re sed on the so-clled counting lemm: if G is k- prtite grph with k-prtition V V k, V = = V k = n, where ll induced iprtite grphs G[V i, V j ] re d, ε)-regulr, then the numer of k-cliques K k in G is d k ) n k ± o)). Frnkl nd Rödl extended Szemerédi s regulrity lemm to -grphs nd Ngle nd Rödl estlished n ccompnying -grph counting lemm nlogous to the grph counting lemm oe. In this pper, we proide new proof of the -grph counting lemm. Key words: Szemerédi s regulrity lemm, hypergrph regulrity lemm, counting lemm 99 MSC: primry 05C65, secondry 05C5, 05D05 Dedicted to Professor Miklós Simonoits on the occsion of his 60th irthdy Corresponding uthor. Emil ddresses: ngle@unr.edu Brendn Ngle), rodl@mthcs.emory.edu Vojtěch Rödl), schcht@informtik.hu-erlin.de Mthis Schcht). Prtilly supported y NSF grnt DMS Prtilly supported y NSF grnt DMS Supported y DFG grnt SCHA 6/-. Preprint sumitted to Discrete Mthemtics 6 April 006

2 Introduction Szemerédi s regulrity lemm [0] is powerful tool in comintorics with mny pplictions in extreml grph theory, comintoril numer theory, nd theoreticl computer science see, e.g., the excellent sureys [8,9] for some of these pplictions). The lemm sserts tht ll lrge grphs cn e decomposed into constntly mny edge-disjoint, iprtite sugrphs, lmost ll of which ehe rndom-like see Theorem elow). The rod pplicility of Szemerédi s lemm to grph prolems suggests tht regulrity lemm for hypergrphs might render mny pplictions. Frnkl nd Rödl [] estlished such n extension, herefter clled the FR-Lemm, of the regulrity lemm to -grphs or -uniform hypergrphs. A -uniform hypergrph H on the ertex set V is fmily of -element susets of V, i.e., H ) V. Note tht we identify hypergrphs with their edge set nd we write V H) for the ertex set.) The FR-lemm gurntees tht ny lrge - grph dmits decomposition into constntly mny edge-disjoint, triprtite susystems, lmost ll of which ehe rndom-like. Applictions of the FR-lemm to -grphs cn e found in [,4 6,0,,5,6,8,9]. Most of the pplictions of the -grph regulrity lemm re sed on structurl counterprt, the so-clled -grph counting lemm, which ws first otined y the first two uthors []. As cogent exmple, the counting lemm, working within the frmework of the FR-lemm, gies new proof of Szemerédi s theorem for rithmetic progressions of length four see []) nd its multidimensionl ersion restricted to four points see [9]). In this note we gie n lterntie proof of the -grph counting lemm, Theorem 5. This result ws originlly otined y the first two uthors [] nd follows lso from the work of Peng, Skokn nd the second uthor [4]. In this ltter reference, the uthors show tht hypergrph regulrity, defined precisely in Definition, is suitly presered on complete underlying sugrphs, which then implies the counting lemm.) The proof presented here is sustntilly different. It is sed on Szemerédi s regulrity lemm nd is somewht simpler thn the erlier proofs. The sttement of Theorem 5 requires some nottion nd we egin y stting Szemerédi s regulrity lemm precisely.. Szemerédi s regulrity lemm In this pper we write x = y ± ξ for rels x nd y nd some positie ξ > 0 for the inequlities y ξ x y + ξ. Szemerédi s lemm piots on the concept of n ε-regulr pir. Let iprtite grph B e gien with iprtition X Y. e sy the pir X, Y ) is d, ε)-regulr if for ll X X nd Y Y where

3 X > ε X nd Y > ε Y, we he d B X, Y ) = d ± ε where d B X, Y ) = EB[X, Y ]) X Y is the density of the iprtite sugrph B[X, Y ] of B induced on X Y. e sy the pir X, Y ) is ε-regulr if it is d, ε)- regulr for some d. In this pper, we use well-known rint of Szemerédi s regulrity lemm for k-prtite grphs G, nd therefore present Szemerédi s lemm in this context. Let k-prtite grph G e gien with k-prtition V = V G) = V... V k. e sy refining prtition i... t i = V i, i k, is t-equitle if i... t i i +. e sy t-equitle prtition i... t i = V i, i k, is ε-regulr if for ll i < j k, ll ut εt pirs, i j ),, t, re ε-regulr. Szemerédi s regulrity lemm for k-prtite grphs) cn then e stted s follows. Theorem Szemerédi s regulrity lemm) Let integer k nd ε > 0 e gien. There exist positie integers N 0 = N 0 k, ε) nd T 0 = T 0 k, ε) such tht ny k-prtite grph G on the ertex set V = V V k with V,..., V k N 0, dmits n ε-regulr nd t-equitle prtition i... t i = V i for i k, where t T 0. Centrl to mny pplictions of Szemerédi s regulrity lemm is the ssertion tht ny sugrph F of constnt size my e emedded into n ppropritely gien collection of dense nd regulr pirs from n ε-regulr nd t-equitle prtition. This osertion is due to the counting lemm for grphs. For grph G, we denote y K s ) G) the set of ll s-tuples from V G) spnning cliques K s ) in G. Fct Counting lemm) For eery integer s nd constnts d > 0 nd γ > 0 there exists ε > 0 so tht wheneer G is n s-prtite grph with ertex prtition V V s stisfying tht ll induced iprtite grphs G[V i, V j ], i < j s, re d, ε)-regulr nd V = = V s = n for sufficiently lrge n, then K s ) G) = d s ) n s ± γ).. The counting lemm for -grphs In this section we introduce the notion of regulr -grphs nd stte the - grph counting lemm. e omit formultion of the FR-Lemm since its There re other k-prtite formultions of Szemerédi s regulrity lemm. A possily more common formultion would define t-equitle prtitions s 0 i i t i = V i, i t, where 0 i < t nd i = = t i 0 i, i t, is often referred to s grge clss). Then ε-regulr, t-equitle prtitions would e defined otherwise the sme s we did for Theorem ; for ech i < j k, ll ut εt pirs, i j ),, t, re ε-regulr. These two notions of t- equitle ε-regulr prtitions re the equilent, howeer, up to slight chnge in ε.

4 formultion is somewht technicl nd, in fct, is not needed to stte the corresponding counting lemm. The following definition generlizes the notion of regulr grphs to regulr -grphs. Definition δ, r)-regulrity) Let positie integer r nd constnts d 0 nd δ > 0 e gien long with -grph H nd -prtite grph P = P P P. e sy tht H is d, δ, r)-regulr with respect to P if for ny fmily Q = {Q,..., Q r } of r sugrphs of P with where r K ) i= Q i ) > δ K ) P ) we he d H Q) d < δ r H i= K) Q i) d H Q) = r if r i= K) Q i= K ) Q i ) > 0, i) 0 otherwise. is the density of H on Q. e sy H is δ, r)-regulr with respect to P if it is d, δ, r)-regulr with respect to P for some d 0. In most contexts where H is d, δ, r)-regulr w.r.t. P, we ctully he H K ) P ). This ssumption, howeer, is not needed to stte Definition. Moreoer, we note tht Definition llows some memers Q i of Q to e empty. hile Szemerédi s regulrity lemm decomposes the ertex set of grph, the -grph regulrity lemm prtitions not only the ertex set, ut lso prtitions the set of ll pirs etween ny two such ertex clsses into edge-disjoint iprtite grphs. In tht enironment, the concept corresponding to n ε-regulr pir is tht of Definition, where the three iprtite grphs P, P, nd P re lso regulr in the sense of Szemerédi). Consequently, corresponding generliztion of Fct tkes plce in the following enironment. Setup 4 Let positie integers k, r nd n nd positie constnts d, δ, d nd δ e gien. Suppose ) V = V... V k, V =... = V k = n, is prtition of ertex set V. ) P = i<j k P ij is k-prtite grph, with ertex set V nd k-prtition oe, where ll P ij = P [V i, V j ], i < j k, re d, δ )-regulr. ) H = h<i<j k H K P ) is k-prtite -grph, with ertex set V nd k-prtition oe, where ll H = H[V h, V i, V j ], h < i < j k, re d, δ, r)-regulr with respect to P hi P ij P hj. The counting lemm estimtes the numer of hypercliques, i.e., complete - grphs, K ) k in H. e denote y K ) k H) the set of ll k-tuples from V H) spnning hypercliques K ) k in H. Theorem 5 Counting lemm []) Let k e n integer. For eery 4

5 γ > 0 nd d > 0 there exists δ > 0 so tht for ll d > 0 there exist integer r nd δ > 0 nd n sufficiently lrge so tht with these constnts, if H nd P re s in Setup 4, then K ) k H) = d k ) d k ) n k ± γ). Proing Theorem 5 is the content of this pper. The first proof of Theorem 5 ppered in [] nd nother proof y Peng, Skokn, nd one of the uthors ws gien in [4]. The proof we present here is shorter thn the preious ones nd we eliee it is lso simpler. e present our proof in Section nd conclude this introduction with the following remrks. The min prolem of proing Theorem 5 is working with the gien quntifiction of constnts: γ, d, δ : d, δ, r. This quntifiction, consistent with the output of the -grph regulrity lemm, llows for the grph P to e reltiely sprse compred to δ, the mesure of regulrity of the -grph H. If the quntifiction of constnts were llowed s γ, d, d, δ = δ, then such dense ersion of Theorem 5 is simpler to proe nd ws proed in [7]. In the present pper, we use Szemerédi s regulrity lemm, Theorem, to oercome those difficulties rising from the quntifiction of constnts in Theorem 5. Recently Gowers [,] deeloped regulrity lemm nd corresponding counting lemm for l-grphs for generl l. The pproch in [,] is different nd, e.g., for l = the notion of -grph regulrity there differs from tht in Definition. A regulrity lemm for l-grphs l ) extending the notion of δ, r)-regulrity ws proed y Rödl nd Skokn [7] nd the current uthors [] proed n ccompnying l-grph counting lemm for tht regulrity lemm. The proof of the generl counting lemm in [] ws inspired y the min ide presented here, i.e., it uses the regulrity lemm for l-grphs to oercome difficulties, which re similr to those indicted in the preious remrk. Acknowledgements The uthors would like to thnk the nonymous referees for their helpful suggestions. 5

6 Proof of the -grph counting lemm It ws shown in [] tht the full sttement of Theorem 5 cn e deduced from just the lower ound. Hence it suffices to proe the lower ound of Theorem 5 only. Our proof of Theorem 5 proceeds y induction on k. The se cse k = is triil. Indeed, y Definition, H = H hs reltie) density d ± δ with respect to P = P P P. Fct implies tht with δ γ) K ) P ) = d n ± γ/) nd the lower ound of Theorem 5 for k = then follows from δ γ. To proceed to the induction step, we ssume tht Theorem 5 holds for k. Reclling the quntifiction of Theorem 5, which is γ, d, d : d, δ, r, we my ssume tht k, γ, d δ min{δ, d } δ, r n ) holds. Then for gien grph P nd -grph H s in Setup 4, we show K ) k H) ) dk d k ) n k γ). e now refine the hierrchy in ) nd introduce some further uxiliry constnts. Let ε 0 > 0 nd integer r > 0 e chosen so tht oth ε 0, /r min{d, δ }. Let T 0 = T 0 k, ε 0 ) e the constnt gurnteed y Szemerédi s regulrity lemm, Theorem. e choose δ > 0 so smll nd integers r nd n so lrge which complies with the quntifiction of Theorem 5) tht the hierrchy ) extends to k, γ, d δ min{δ, d } ε 0, r, T 0 δ, r n. ) Before going into the precise detils of the induction step, we first gie n informl description of the proof.. Outline of the induction step The so-clled link grphs of H ply centrl rôle in our proof of the induction step. In the context of Setup 4, consider ertex V nd fix i < j k. The i, j)-link grph L ij is defined s L ij = {{ i, j } P ij : {, i, j } H} Note tht L ij hs ertex set N P i) N P j) where, for exmple, N P i) is the P i -neighorhood of the ertex. Note tht L ij is sugrph of P ij, 6

7 nd the link grph L of is then set s L = i<j k L ij. Note tht L is k )-prtite grph.) A nturl plce to consider pplying the induction hypothesis on the counting lemm is to enumerte cliques K ) k in the k )-prtite hypergrph H K ) L ) with the k )-prtite grph L ), where V is typicl ertex. Indeed, clique K ) k in H K) L ) corresponds to clique K ) k in H contining the ertex.) For this, one would need to check tht the hypothesis of the counting lemm is met for k )) y H K ) L ) nd L replcing H nd P, s in Setup 4). Unfortuntely, this won t often e the cse. Indeed, one my show tht while the density of the iprtite grphs L ij for most V ), i < j k, is out d d, the regulrity of L ij depends on δ. As we see in ), δ d d, nd to pply the induction hypothesis, we would need it the other wy round. The min ide of our proof is to pply the Szemerédi regulrity lemm, Theorem, to the link grphs L, i.e., we regulrize the links. ith ε 0 d d cf. )), we will regulrize ech L to otin ε 0 -regulr prtition P gien y V i =,i,i t, i k, where t T 0 for the constnt T 0 ppering in ). e will then show tht for ech i < j k, for most V, most of the pirs,i,,j,, t, will he density in L ij close to d d see prt i ) of Clim 7). Of course, most of these pirs,i,,j re ε 0 -regulr where ε 0 d d.) Showing this will inole using the d, δ, r)-regulrity of H ij w.r.t. P i P j P ij nd the choice r T 0. e will then show tht for ll h < i < j k, for most V, most triples,h,,i, c,j,,, c t, will stisfy tht H K ) L ) is d, δ /0, r )-regulr w.r.t. L [,h,,i, c,j ] see prt ii ) of Clim 7). Showing this will inole using the d, δ, r)-regulrity of H w.r.t. P hi P hj P ij nd the choice r mx{r, T 0 }. From the two osertions oe, we then infer tht for most V, most k )-prtite grphs L [,,...,,k k ],,..., k t, nd corresponding -grphs H K ) L [,,...,,k k ]) stisfy the hypothesis for k )) of the counting lemm. Tht is, fter the djustment of regulrizing the links, we re in position of using the induction hypothesis within the pieces).) e then use the induction hypothesis to count the cliques K ) k in H K ) L [,,...,,k k ]). e then dd oer ll suitle choices of indices,..., k t nd then dd oer ll suitle choices of ertices V. e now formlize the detils sketched oe. where P ij = P ij [N P i), N P j)] is the sugrph of P ij induced on the neighorhoods N P i) nd N P j). 7

8 . Trnsersls nd their properties Let the constnts e fixed s in ) nd k-prtite grph P nd -grph H e gien s in Setup 4. e first regulrize the link grphs. For eery ertex V, we pply Szemerédi s regulrity lemm, Theorem with ε 0, to the k )-prtite link grph L to otin n ε 0 -regulr nd t -equitle prtition P of V L ), where t T 0 see )). In other words, P refines the prtition N P ) N P k) = V L ), i.e., we otin,i,i t = N P i) for i =,..., k, where for ll pirs i < j k ll ut t most ε 0 t pirs, ) [t ] [t ] stisfy tht L ij,,j ] is ε 0 -regulr. [,i For fixed V nd fixed k )-ector =,..., k ) [t ] [t ] = [t ] k we denote y L ) the sugrph of L induced on the sets,...,,k, i.e.,, k L ) = i<j k L ij [ ] [,i i,,j j = L, ],...,,k k. ) Similrly, we define for ll h < i < j k nd h, i, j ) [t ] L [ h, i, j ] = L hi [,h h,,i i ] L ij [,i i,,j ] L hj j [,h h,,j j ]. 4) Moreoer, we set H ) to e equl to the -grph H induced on the tringles of L ), i.e., H ) = H K ) L ) ) = where H ) = H K ) L [ h, i, j ]). h<i<j k H ), 5) e refer to the ojects H ) nd L ) s trnsersls of the prtition P see Figure ). Note tht s L ws regulrized, we infer tht ll ut ε 0 k t k ectors =,..., k ) [t ] k stisfy tht ll ) k iprtite grphs L ij [,i i,,j j ], i < j k, re ε 0 -regulr. It follows directly from the definitions in ) nd 5) tht K ) k H) = V [t ] k K ) k H )). 6) In our proof of the induction step we will use the following well-known fct out the size of typicl neighorhoods in δ -regulr grphs see, e.g., [8, Fct ]). 8

9 V, V, V,i i V i,j j V j,k k V k Fig.. A trnsersl of the prtition P. Fct 6 For ll ut kδ n ertices V, we he N P i) = d ± δ )n, for ll i k. For future reference, we set V = { V : N P i) = d ± δ )n, for ll i k} 7) so tht Fct 6 implies V kδ )n. The following clim is the key osertion for the proof of Theorem 5. hile technicl looking, prt i ) of Clim 7 follows from stndrd rguments, which we present in Section 4. The proof of prt ii ) is gien in Section 5. Clim 7 For ll ut δ /4 n ertices V see 7)), ll ut δ /0 k t k ectors =,..., k ) [t ] k yield trnsersls L ) nd H ) stisfying tht i ) for ll i < j k the iprtite grphs L ij [,i i,,j ] he density d d ± δ /4 ) nd due to regulriztion) re ε 0 -regulr, ii ) for ll h < i < j k the -prtite -grph H ) is d, δ /0 regulr with respect to the -prtite grphs L in ) recll the nottion in 4)). j, r )- [ h, i, j ], where r is gien Let V typ denote the set of typicl ertices V to which Fct 6 nd Clim 7 refer. For ech V typ, let [t ] k typ denote the set of typicl ectors [t ] k to which Clim 7 refers. 9

10 . The induction step e conclude from Fct 6 nd Clim 7 oe tht for ny ertex V typ nd ny [t ] k typ, the trnsersls H ) nd L ), stisfy the hypothesis of Setup 4 with the constnts k, d, δ /0, d d, ε 0, r nd d n/t. Indeed, s in Setup 4, osere tht H ) replces H, L ) replces P, k replces k, d remins d, δ /0 replces δ, d d replces d, ε 0 replces δ nd d n/t replces n. e ll tke γ/ to replce γ) Due to the hierrchy of the constnts in ), we my ssume tht k, γ, d δ /0 min{δ /0, d d } ε 0, r t d n. 8) As such, for fixed V typ nd =,..., k ) [t ] k typ, we my pply the induction hypothesis to the trnsersls H ) nd L ) nd infer K ) k H )) d k ) d d ) k = d k ) d k ) n k t k ) d n γ t ) ) k γ ). 9) Consequently, y 6) we he K ) k H) = V [t ] k K ) k H )) 9) d k ) d k ) n k γ ) V typ [t ] k typ. t k By Fct 6 nd Clim 7, V typ δ /4 kδ )n > δ /4 )n nd [t ] k typ k δ /0 )t k for eery V typ. Hence we conclude due to the hierrchy in 8)) tht K ) k H) d k ) d k ) n k γ ) δ /4 ) k δ /0 ) d k ) d k ) n k γ). This concludes our proof of Theorem 5. 0

11 Proof of Clim 7 In this section, we outline our strtegies for proing prts i ) nd ii ) of Clim 7. To egin, we find the following nottion helpful to discuss Clim 7 nd use it in the reminder of this pper. Definition 8 Fix V. For fixed i < j k nd h < i, set L ij good ) = {, ) [t ] : L ij [,i,,j ] is d, ε 0 )-regulr for d = d d ± δ /4 )}, L good ) = {,, c) [t ] :, ) L hi good ),, c) Lij good ),, c) Lhj good )}, H good ) = {,, c) [t ] : H is d, δ /0, r )-regulr w.r.t L [,, c]}, where L [,, c] ws defined in 4). Finlly, set L good ) = { [t ] k : i, j ) L ij good ) for ll i < j k}, H good ) = { [t ] k : h, i, j ) H good ) for ll h < i < j k}. e lso define corresponding d sets nd fix L ij d ) = [t ] \ L ij good ), L d ) = [t ] \ L good ), H d ) = [t ] \ H good ), L d ) = [t ] k \ L good ), nd H d ) = [t ] k \ H good ). In the nottion oe, Clim 7 sserts tht ll ut δ /4 n ertices V see 7)) stisfy L d ) + H d ) δ /0 t k. e consider the sum on the left hnd side of the inequlity oe. Osere tht L d ) + H d ) = L d ) + H d ) L good ) + H d ) L d ) L d ) + H d ) L good ). Moreoer, osere tht nd L d ) t k H d ) L good ) t k 4 L ij i<j k d ) H d h<i<j k ) L good ) hold for ll V. e my therefore gie reformultions of prts i ) nd ii ) from Clim 7 in the following form. Proposition 9 Clim 7 prt i )) Let P nd H stisfy Setup 4 with constnts s in ). Then ll ut k δ / n ertices V see 7)) stisfy tht ) δ/4 t for ll i < j k. L ij d

12 Proposition 0 Clim 7 prt ii )) Let P nd H stisfy Setup 4 with constnts s in ). Then ll ut k δ /4 n ertices V see 7)) stisfy tht ) H ) < δ/0 t for ll h < i < j k. L good d Propositions 9 nd 0 together imply tht ll ut k δ / n + k δ /4 n δ /4 ertices V stisfy L d ) + H d ) L good ) t k L ij d ) + t k 4 i<j k s promised y Clim 7. 6δ /4 H d h<i<j k ) k t k + δ /0 ) L good ) n ) k t k δ /0 k t k, e gie the proofs of Proposition 9 nd Proposition 0 in Section 4 nd Section 5, respectiely. 4 Proof of Proposition 9 Let P nd H e gien s in Setup 4 where the constnts stisfy ). Moreoer, let {P } V e the fmily of ε 0 -regulr, t -equitle prtitions otined in Section.. e proe tht ll ut k δ / n ertices V see 7)) stisfy L ij d ) δ/4 t for ll i < j k. Let us clrify this gol. Fix i < j k. Since P is ε 0 -regulr for eery V, t most ε 0 t δ /4 t pirs,,j ),, t, cn e irregulr. Hence we only he to erify the,i density ssertion of Proposition 9, nmely, for ll ut δ / n ertices V, d L ij,i holds for ll ut δ /4 t pirs,i,,j ). e egin with the following definition.,,j ) = d d ± δ /4 ), Definition Let L P e iprtite grphs with iprtition U U nd let d, δ > 0 nd integer r e gien. e sy L is d, δ, r)-regulr with respect to P if eery fmily B = {B,..., B r } of r induced sugrphs B i P stisfying r s= B s > δ P lso stisfies L r s= B s = d ± δ) r s= B s. The following fct ppered in slightly different lnguge) in [, Clim A] see lso []). It sserts tht for H nd P s in Setup 4, most ertices V stisfy tht their links L ij, i j k, re regulr in the sense of Definition.

13 Fct most links re d, δ /, r)-regulr) Let k, d, δ, d nd r e gien s in ). Then for H nd P re s in Setup 4, ll ut k δ / n ertices V see 7)) stisfy tht for ll i < j k, L ij is d, δ /, r)-regulr with respect to P ij = P [N P i), N P j)]. Fct is essentilly the sme s Clim A from []. For completeness, we sketch proof of Fct t the end of this section. As in Fct, we sy tht ertex V is good ertex if for ll i < j k, L ij is d, δ /, r)-regulr with respect to P ij. Let V good = V good k, d, δ, d, δ, r) e the set of ll good ertices V. PROOF of Proposition 9. Fct ensures us tht lmost eery ertex V is good ertex. Now, fix i < j k. The key osertion is tht eery good ertex V good stisfies tht ll ut δ /4 t pirs,i,,j,, t, he density d d ± δ /4 ). Indeed, let V good ut suppose {,i,,j )},) I is collection of pirs, ech with density, sy, smller thn d d δ /4 ), such tht I δ /4 t. e clim the fmily B = {P ij [,i,,j ]:, ) I} contrdicts the d, δ /, r)- regulrity of L ij with respect to P ij. Note tht ) gies tht r T0 fmily of r induced sugrphs of P ij fmily of r induced sugrphs of P ij nd Lij,) I P ij,) I [,i,,j P ij t I = B. The set B is therefore = P [N P i), N P j)]. e clim B is stisfying [,i,,j ] > δ / P ij ] < d δ / ) P ij,) I 0) [,i,,j ]. ) Once 0) nd ) re estlished, we see tht B contrdicts the d, δ /, r)- regulrity of L ij with respect to P ij. This will proe Proposition 9. e first erify 0). Osere tht,) I P ij [,i,,j ] =,) I P ij [,i,,j Fix, ) I. Recll tht δ /T 0 /t in ) nd since V,i = N P i) ± = d ± δ ) n, t t ]. )

14 recll 7)). Consequently, the d, δ )-regulrity of P ij implies tht P ij [,i,,j ] = d ± δ ),i,j = d ± δ ) d ± δ ) n ) = d ± δ ) n. t t ) The d, δ )-regulrity of P ij lso implies reclling V cf. 7)) P ij = d ± δ ) d ± δ )n) = d ± δ ) n. 4) Consequently, with I δ /4 t, ), ) nd 4) estlish 0). Osere tht ) is equilent to,) I L ij [,i,,j Fix, ) I. Our ssumption is tht L ij [,i which, with ), implies L ij [,i ] < d δ / ),) I,,j,,j ] < d d δ /4 ] < d δ /4 ) δ d ) P ij [,i,,j ),i,j P ij < d δ / ) P ij [,i [,i,,j,,j ] ]. 5) ] 6) where the lst inequlity follows from δ d, δ in ). As 6) holds for ech, ) I, 5) follows. PROOF of Fct. It suffices to consider just the cse k =, for which we proe ll ut δ / n ertices V see 7)) stisfy tht L is d, δ /, r)- regulr w.r.t. P. e note tht while the constnts d, δ, d, δ nd r stisfy the hierrchy in ) due to the quntifiction of the counting lemm), ll tht is required to enle the present sketch is tht 0 < δ = δ d ) d is sufficiently smll. For ech fixed ertex V see 7)) for which L is not d, δ /, r)-regulr w.r.t. P, fix fmily B = {B,..., B r } of r induced sugrphs B s P, s r, for which r B s > δ / P 7) s= 4

15 ut for which either r L B s < d δ / r r ) B s or L B s > d + δ / r ) B s. s= s= s= s= Let V e the set of such ertices V for which the first condition holds nd let V + e the set of such ertices V for which the second condition holds. e clim V < δ / n nd V + < δ / n. The proofs of these two inequlities re symmetric, so w.l.o.g., we proe only the first. Assume, on the contrry, tht V δ / n. e show V leds to contrdiction with the d, δ, r)-regulrity of H w.r.t. P P P. In prticulr, we show the set V implies the existence of fmily Q = Q V = {Q,..., Q r } stisfying r K ) Q s ) > δ K ) P P P ) nd d H Q) < d δ. 8) s= Indeed, fix V nd fix s r. Define Q s P respectiely Q s P ) s the set of ll edges of P resp. P ) contining ertex nd define Q s = B s. Set Q s = Q s Q s Q s nd Q s = V Q s. Set Q = {Q,..., Q r }. Note tht nd r K ) Q s ) = r B s 9) s= V s= r H K ) Q s ) = r L B s. 0) s= V s= Note tht the second inequlity of 8) is triil. Indeed, using oth equlities in 9) nd 0) nd the definition of V, we he V r L s= B s < d δ / ) V r s= so tht d H Q) < d δ / < d δ follows. To see the first inequlity of 8), we use 7) to see V r B s > s= V B s = d δ / ) r K ) s= δ / P > δ / d δ ) d δ )n) V Q s ) where the lst inequlity follows from V s in 4) cf. 7)). Then our 5

16 ssumption out V implies δ / d δ ) n V > δ d δ ) n. Since δ d, Fct implies K ) P P P ) /)d n, nd so the first inequlity of 8) follows from 9) nd from δ d d in ). 5 Proof of Proposition 0 e show tht ll ut k δ /4 n ertices V see 7)) stisfy L good ) H d ) < δ/0 t for ll h < i < j k. In the reminder of this pper, we fix h < i < j k. It suffices to proe tht ll ut δ /4 n ertices V stisfy L good ) H d ) < δ/0 t for the fixed indices h < i < j k. Remrk In the reminder of this pper, the indices h < i < j k re fixed. Assume, on the contrry, there exists set A V of size consisting of ertices for which A > δ /4 n ) L good ) H d ) /0 δ t. ) e show tht ) leds to contrdiction to our hypothesis of Setup 4 tht the trid H is d, δ, r)-regulr with respect to P hi P ij P hj. e outline our pproch in the following remrk. Remrk 4 Fix A nd fix,, c) L good ) H d ). Since,, c) ), we ppel to Definitions nd 8 to infer tht there exists fmily H d Q c = {Q c p) : p r }, Q c ut r K ) Q p= p) L c p)) > δ/0 K ) [,, c] see 4)), stisfying L [,, c] ), ) ) dh Q /0 c d δ. 4) In ), we collect witness Q c for ech,, c) L good ) H d ) nd A to crete ig witness Q tht will contrdict the d, δ, r)- regulrity of H with respect to P hi P ij P hj. 6

17 In the process of collecting the witnesses Q c oer,, c) L good ) H d ) nd A, we do not need the entire set A, nd in fct, we need only smll suset thereof. Oer two steps, we refine the set A into two nested susets C B A where the finl suset C produces the ig witness Q promised. 5. Refining the set A e otin the intermedite suset B A using Fct 5 elow. This fct sttes tht from A we my find suset of ertices B, eery pir from which hs the right shred P q -neighorhood, q {h, i, j}. Fct 5 Set f = 8 δ/5. 5) d d Assuming ), there exists set B A of size B = f such tht for ech q {h, i, j} nd for eery distinct ertices u, B, N P qu) N P q) = d ± δ ) n. 6) Fct 5 is not difficult to proe nd ws shown, in slightly different context, in [, pge 55]. For completeness, we proe Fct 5 in Section 5.5. To identify the suset C B, we use the following considertions. Fix B nd set { LH ) =,, c) L good ) H d ): d ) } H Q c < d δ /0, 7) { LH + ) =,, c) L good ) H d ): d ) } H Q c > d + δ /0. 8) Moreoer, we define { } B = B : LH ) L good ) H d ), { B+ = B : LH + ) L good ) H d ) }. Clerly, one of B B = f or B + B = f holds. In our proof, it does not mtter which holds s the cses re symmetric. e ssume, without loss of generlity, tht the former holds nd we fix some set C B B such tht C = f. 9) 7

18 e construct the witness Q from C. Before doing so, howeer, we stte the following fct for future reference. Fct 6 Let C. From ) nd the definition of T 0 see )), we infer δ /0 t L d ) H d ) LH ) t T0 0) ) For ech,, c) LH ), we recll from 4) tht d H Q c < d δ /0, nd so, r H K ) Q p= c p)) < d δ /0 ) r p= K ) Q c p)). ) 5. Constructing the witness ith the set C oe, we proceed to construct the promised witness Q. Define Q = { Q c p): C,,, c) LH ), nd p =,..., r }. ) e ssert Q is the promised fmily witnessing the d, δ, r)-irregulrity of H with respect to P hi P ij P hj. e first clim tht Q hs t most r memers. Indeed, we he s desired. Q ) = r LH ) 0) r ft0 C 5) = 8r T 0 δ /5 d d ) r, Now, s Q hs t most r memers consisting of sugrphs from P hi P ij P hj, the following osertion, Clim 7 nd 8, proide direct contrdiction to the d, δ, r)-regulrity of H with respect to P hi P ij P hj. For tht set K ) Q ) = { K ) Q c p)): C,,, c) LH ), p =,..., r }. Clim 7 K ) Q K ) ) > δ P hi P ij P hj ). Clim 8 H K ) Q ) < d δ ) K ) Q ). Since Clims 7 nd 8 proide contrdiction to the d, δ, r)-regulrity of H with respect to P hi P ij P hj, our proof of Proposition 0 will e complete upon proing these two clims. 8

19 5. Proof of Clim 7 Inclusion-exclusion gies K ) Q ) { ) K C C Q c p)) :,, c) LH ), p =,..., r } { ) K Q c p)) K ) Q c p ) )}, ) where the lst union runs oer ll,, c) LH ),,, c ) LH ), nd p, p =,..., r. e ound the two terms on the right hnd side of ) in the following two fcts. Fct 9 For eery C { ) K Q c p)):,, c) LH ), p =,..., r } δ /0 8 d d 6 n. Fct 0 For ll distinct ertices, C { ) K Q c p)) K ) Q c p ) ) :,, c) LH ),,, c ) LH ), nd p, p =,..., r } 6d 9 n. Fcts 9 nd 0 conclude the proof of Clim 7. PROOF of Clim 7. Applying Fcts 9 nd 0 to ), we otin the lower ound ) K ) Q ) f δ/0 f 8 d d 6 n 6 d 9 n d n fd d δ /0 8f d 6. 8 Inserting the lue f = 8δ /5 /d d ) from 5), we infer the further lower ound K ) Q ) d n δ / 7 d 6 ) δ 4/5 = δ / d n 7 δ /0 d 6 δ/ d n, 4) where the lst inequlity follows from the fct tht δ d from ). On the other hnd, since δ d in ), we conclude from Fct, the counting lemm These two fcts will lso e useful in our proof of Clim 8, s will the inclusionexclusion of ). ) 9

20 for grphs, tht K ) P hi P ij P hj) d n. Compring this inequlity ginst 4) proes Clim 7. Thus, it remins to erify Fcts 9 nd 0. PROOF of Fct 9. Fix ertex C. Osere from ) tht { ) K Q c p)):,, c) LH ) nd p =,..., r } = r,,c) LH ) p= ) K ) Q,,c) LH ) e further estimte 5) y ppeling to Fct. c p)) δ /0 K ) L [,, c] ). 5) Fix,, c) LH ) L good ) see 7)). By the definition of L good ), ech of the three iprtite grphs L hi [,h,,i ], L ij [,i, c,j ], nd L hj [,h, c,j ], is ε 0 -regulr with density d d ± δ /4 ), where ε 0 d d from ). Applying Fct to L [,, c], we therefore conclude K ) L [,, c] ) d d δ /4 d d ) 6,h )),h,i c,j,i c,j d d 6 n 8 t 6) where the lst inequlity follows from the fct tht V see 7)). Applying 6) to 5), we conclude { ) K Q c p)):,, c) LH ) nd p =,..., r } δ/0 8 d d 6 n LH t ) 0) δ/0 8 d d 6 n, s climed. PROOF of Fct 0 Let two distinct ertices nd C e fixed. e use the nottion P hi for the sugrph of P hi induced on N P h, ) N P i, ) where, for exmple, N P h, ) = N P h) N P h ). Define P ij hj nd P 0

21 similrly. Then, { ) K Q c p)) K ) Q c p ) ) :,, c) LH ), 7) ) P hi P ij P hj.,, c ) LH ), nd p, p =,..., r } K ) To ound the right hnd side of 7), we pply Fct to the grph P hi P ij P hj, ut first check tht it is pproprite to do so. To see tht Fct pplies to the grph P hi P hj, we clim tht ech of P hi ij, P hj nd P is d, δ / )-regulr, nd check this ssertion for P hi Recll from 6) tht ech of N P h, ), N P i, ) = d ± δ ) n δ n. Since P hi is d, δ )-regulr, nd since P hi is the sugrph of P hi induced on N P h, ) N P i, ), we he tht P hi inherits d, δ / )-regulrity from P hi. Returning to 7), we pply Fct with δ / d ) to otin K ) P hi P ij P hj ) d N P h, ) N P i, ) N P j, ), from which it follows i 6)) tht K ) P hi P ij P hj ) 6d 9 n. 8) Comining 7) nd 8) proes Fct Proof of Clim 8 The proof of Clim 8 follows lrgely from work of the proof of Clim 7. First, osere tht H K ) Q ) ) C,,c) LH ) < d δ /0 = d δ /0 ) C r H p= ) r C,,c) LH ) p= r,,c) LH ) p= K ) Q c p)) K ) Q { ) K Q c p)) c p))}. As one my show, in fct, P hi inherits δ /d )-regulrity from P hi.

22 Recll tht we sw the right-most sum oe in our inclusion-exclusion of ). In prticulr, we my use ) nd Fct 0 to otin the further upper ound H K ) Q ) < d δ /0 ) K ) Q ) + 6 ) f d 9 n ). As such, we use Fct 9 nd the definition of Q in ) to infer d H Q ) < d δ /0 ) 6 ) f d 9 n + fδ /0 d d 6 n /8 d δ /0 ) + 0 fd δ /0 d Using the lue f = 8δ /5 /d d ) see 5)), we otin further upper ound d H Q ) < d δ /0 ) + 7 δ /0 < d d 6 δ, where the lst inequlity follows from δ d in ). This proes Clim 8. ). 5.5 Proof of Fct 5 The proof depends only on the hypotheses tht the iprtite grphs P h, P i, nd P j re ech d, δ )-regulr nd tht A > δ /4 n s we ssumed in )). In prticulr, the hypothesis in ) will ply no rôle in wht follows. e shll pply Turán s theorem [] to the uxiliry grph Γ = V Γ), EΓ)) whose ertices re gien y V Γ) = A V nd whose edges re gien y { ) } A EΓ) = {, } : N P q, ) = d ± δ ) n, q {h, i, j} where, for ertices, nd index q {h, i, j}, N P q, ) = N P q) N P q )). Indeed, with f = 8δ /5 /d d ) gien in 5), note tht we my tke the desired set B A s the ertex set of ny clique K f in Γ. Suppose, on the contrry, tht Γ contins no cliques K f. Then Turán s theorem ensures EΓ) ) A ) f + o) where o) 0 s A. Since A > δ /4 n, where ) ensures n my e tken s lrge s we need, we infer ) A ) EΓ) ) A ). 9) f ) 8f

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