Extremal problems on ordered and convex geometric hypergraphs

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1 Extemal poblems on odeed and convex geometic hypegaphs Zoltán Füedi Tao Jiang Alexand Kostochka Dhuv Mubayi Jacques Vestaëte July 16, 2018 Abstact An odeed hypegaph is a hypegaph whose vetex set is linealy odeed, and a convex geometic hypegaph is a hypegaph whose vetex set is cyclically odeed. Extemal poblems fo odeed and convex geometic gaphs have a ich histoy with applications to a vaiety of poblems in combinatoial geomety. In this pape, we conside analogous extemal poblems fo unifom hypegaphs, and discove a geneal patitioning phenomenon which allows us to detemine the ode of magnitude of the extemal function fo vaious odeed and convex geometic hypegaphs. A special case is the odeed n-vetex -gaph F consisting of two disjoint sets e and f whose vetices altenate in the odeing. We show that fo all n 2 + 1, the maximum numbe of edges in an odeed n-vetex -gaph not containing F is exactly ( ) n ( n This could be consideed as an odeed vesion of the Edős-Ko-Rado Theoem, and genealizes ealie esults of Capoyleas and Pach and Aonov-Dujmovič-Moin-Ooms-da Silveia. ). 1 Intoduction An odeed gaph is a gaph togethe with a linea odeing of its vetex set. Extemal poblems fo odeed gaphs have a long histoy, and wee studied extensively in papes by Pach and Tados [14], Tados [18] and Koándi, Tados, Tomon and Weidet [11]. Let ex (n, F ) denote the maximum numbe of edges in an n-vetex odeed gaph that does not contain the odeed gaph F. This extemal poblem is phased in [11] in tems of patten-avoiding matices. Macus and Tados [13] showed that if the fobidden patten is a pemutation matix, then the answe is in fact linea in n, and theeby solved the Stanley-Wilf Conjectue, as well as a numbe of othe well-known open poblems. A cental open poblem in the aeas was posed by Pach and Tados [14], in the fom of the following conjectue: Conjectue A. Let F be an odeed acyclic gaph with inteval chomatic numbe two. ex (n, F ) = O(n polylog n). Then Reseach suppoted by gant K fom the National Reseach, Development and Innovation Office NKFIH and by the Simons Foundation Collaboation gant # Reseach patially suppoted by National Science Foundation awad DMS Reseach suppoted in pat by NSF gant DMS and by gants A and of the Russian Foundation fo Basic Reseach. Reseach patially suppoted by NSF awads DMS and Reseach suppoted by NSF awad DMS

2 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 2 In suppot of Conjectue A, Koándi, Tados, Tomon and Weidet [11] poved fo a wide class of foests F that ex (n, F ) = n 1+o(1). This conjectue is elated to a question of Baß in the context of convex geometic gaphs. A convex geometic gaph is a gaph togethe with a cyclic odeing of its vetex set. Given a convex geometic gaph F, let ex (n, F ) denote the maximum numbe of edges in an n-vetex convex geometic gaph that does not contain F. Extemal poblems fo geometic gaphs have a faily long histoy, going back to theoems on disjoint line segments [10, 17, 12], and moe ecent esults on cossing matchings [3, 5]. Motivated by the famous Edős unit distance poblem, the fist autho [7] showed that the maximum numbe of unit distances between points of a convex n-gon is O(n log n). In the vein of Conjectue A, Baß [2] asked fo the detemination of all acyclic gaphs F such that ex (n, F ) is linea in n, and this poblem emains open. In this pape, we study extemal poblems fo odeed and convex geometic unifom hypegaphs. An odeed -gaph is an -unifom hypegaph whose vetex set is linealy odeed. A convex geometic -gaph is an -unifom hypegaph whose vetex set is cyclically odeed. We denote by ex (n, F ) the maximum numbe of edges in an n-vetex odeed -gaph that does not contain F, and let ex(n, F ) denote the usual (unodeed) extemal function. Similaly we wite ex (n, F ) in the convex geometic hypegaph setting. As is the case fo convex geometic gaphs, the extemal poblems fo convex geometic hypegaphs ae fequently motivated by poblems in discete geomety [4, 15, 2, 1]. Instances of the extemal poblem fo two disjoint tiangles in the convex geometic setting ae connected to the well-known tiangle-emoval poblem [9]. In [8] we show that cetain types of paths in the convex geometic setting give the cuent best bounds fo the notoious extemal poblem fo tight paths in unifom hypegaphs. One of the goals of this pape is to show similaities and diffeences in solutions of an extemal poblem in linealy odeed and cyclically odeed settings. 2 Results 2.1 A splitting theoem Given subsets A, B of an odeed set, wite A < B to mean that a < b fo each a A and b B. Fo k 2, an odeed -gaph has inteval chomatic numbe k if its vetex set can be patitioned into k sets A 1 < A 2 < < A k such that evey edge has at most one vetex in each A i. Of paticula inteest to us is the case k =, when the sets A i give an -patition of the -gaph. Let z (n, F ) denote the maximum numbe of edges in an n-vetex odeed -gaph of inteval chomatic numbe that does not contain the odeed gaph F. Pach and Tados [14] showed that any n-vetex odeed gaph may be witten as a union of at most log n edge disjoint subgaphs each of whose components is a gaph of inteval chomatic numbe two, and deduced fo evey odeed gaph F that ex (n, F ) = O(z (n, F ) log n). Ou fist esult genealizes thei esult to hypegaphs. Theoem 2.1. Fix c 1 1 and an odeed -gaph F with z (n, F ) = Ω(n c ). Then ex (n, F ) = { O(z (n, F ) log n) if c = 1 O(z (n, F )) if c > 1.

3 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 3 We will give a shot self-contained poof of Theoem 2.1, although it also follows quickly fom ou next esult, which is the main new ingedient in this wok. Definition 1. An odeed -gaph F is a split hypegaph if thee is a patition of V (F ) into intevals X 1 < X 2 < < X 1 and thee exists i [ 1] such that evey edge of F has two vetices in X i and one vetex in evey X j fo j i. Fo instance, evey -gaph of inteval chomatic numbe is a split hypegaph. We wite e(h) fo the numbe of edges in a hypegaph H, v(h) = e H e and d(h) = e(h)/v(h) 1 fo the codegee density of H. Theoem 2.2. Fo evey 3 thee exists c = c > 0 such that evey odeed -gaph H contains a split subgaph G with d(g) c d(h). In the next section, we descibe an application of Theoems 2.1 and 2.2 to extemal poblems fo odeed -gaphs, which demonstates that loss of the facto log n between ex (n, F ) and z (n, F ) is sometimes necessay. This example will also eveal a discepancy between the extemal functions fo an odeed -gaph in the odeed setting vesus the convex geometic setting. 2.2 Cossing paths A tight k-path is an -gaph whose edges have the fom {v i, v i+1,..., v i+ 1 } fo 0 i < k. Typically, we list the vetices v 0 v 1... v k+ 2 in a tight k-path. We conside odeed tight paths to which Theoem 2.2 applies, and fo which we obtain the exact odeed extemal function in a numbe of cases. We let < denote the undelying odeing of the vetices of an odeed o convex geometic hypegaph. Definition 2 (Cossing paths). An -unifom cossing k-path CP k is a tight k-path v 0v 1... v +k 2 with the odeing (i) v 0 < v 1 < v 2 < < v 1, (ii) v j < v j+ < v j+2 < < v j+1 fo j < 1 and (iii) v 1 < v 2 1 < v 3 1 <. An example of an odeed CP5 2 (Figue 1) and of a convex geometic CP 7 2 and CP 5 3 shown below. (Figue 2) ae Ou fist esult detemines the ode of magnitude of the extemal function fo cossing paths in the odeed setting, and the exact extemal function fo shot cossing paths. We note that thee ae vey few exact esults known fo odeed gaphs o hypegaphs. Theoem 2.3. Let k 1, 2 and n + k. Then {( n ) ( ex (n, CPk ) = n k+1 ) fo k + 1 Θ(n 1 log n) fo k + 2. Theoem 2.3 fo k + 2 shows that the log n facto in Theoem 2.2 is necessay, as we shall see fo all k, 2 that z (n, CP k ) = O(n 1 ).

4 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 4 Figue 1: Odeed CP 2 5 Figue 2: Convex Geometic CP 2 7 and CP 3 5 In the convex geometic setting, Baß, Káolyi and Valt [3] poved that ex (n, CP3 2 ) = 2n 3 fo n 3. We genealize this to CPk fo > 2 and k > 3 in the following theoem: Theoem 2.4. Let k 1, 2 and n Then Θ(n 1 ) fo k 2 1 ex (n, CPk ) = ( n ( ) n ) fo k = + 1 Θ(n 1 log n) fo k 2. This eveals a discepancy between the odeed setting and the convex geometic setting: in the convex geometic setting, cossing paths of length up to 2 1 have extemal function of ode n 1, wheeas this phenomenon only occus fo cossing paths of length up to +1 in the odeed setting. In fact, we know that ex (n, CPk ) = ex (n, CPk ) iff k {1, + 1}. The poofs of Theoems 2.3 and 2.4 ely substantially on Theoems 2.1 and 2.2. Theoem 2.3 has a simple coollay fo cossing matchings: a cossing matching CM consists of two disjoint -sets {v 0, v 2, v 4,..., v 2 2 } and {v 1, v 3,..., v 2 1 } such that v 0 < v 1 < v 2 < < v 2 1. In this way, Theoem 2.3 could be viewed as an odeed vesion of the Edős-Ko-Rado Theoem. Aonov, Dujmovič, Moin, Ooms and da Silveia [1] showed that ex (n, CM 3 ) = Θ(n 2 ) and Capoyleas and Pach [5] poved an exact esult fo the convext geometic gaph compising k paiwise cossing line segments. Stating with the simple obsevation that the odeed cossing path CP+1 contains CM, and also that ex (n, CM ) = ex (n, CM ), we obtain the following

5 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 5 coollay to Theoem 2.3 fo k = + 1: Coollay 2.5. Fo n > > 1, ex (n, CM ) = ex (n, CM ) = ( ) ( n n ). We shall see that the same convex geometic -gaph which does not contain CP+1 used to pove Theoem 2.3 also does not contain CM, which establishes the equality in the coollay. 3 Poof of Theoem 2.1 In this section, we suppose that the undelying set (the set of vetices) of an odeed hypegaph is [n]. An inteval is a set of consecutive vetices in the odeing. Given a set of intevals I 1 < I 2 < < I a box B(I 1,...,, I ) is a set of (odeed) -sets {x 1, x 2,..., x } such that x i I i. We say that a box B(I 1,...,, I ) is coveed by (o contained in) the box B(J 1,...,, J ) if I t J t fo all t []. A weighted -unifom hypegaph on a set X is a function ω : ( ) X [0, ). Fo a family F, let w(f) := F F w(f ). Theoem 2.1 follows fom the following moe geneal esult. Theoem 3.1. Let c 1 1 and let ω : ( ) [n] [0, ) be a weighted -unifom hypegaph. Suppose that thee is some A > 0 such that w(b) Al c fo evey box B(I 1,...,, I ) with I 1 = = I = l. Then w (( )) { [n] CAn < 1 log n if c = 1 CAn c if c > 1, whee the C depends only on in the fist case and only on and c in the second case. Poof. Since the statement is monotone, to avoid ceilings and floos, fo easie pesentation we suppose that n = g fo some intege g 1. Define a system of intevals I 1,..., I g and systems of boxes J 1,..., J g as follows. The system I t is obtained by splitting [n] into t equal intevals. So I t = t and each membe of it has length n/ t. Fo any family of (disjoint) intevals I, let B (I) (o just B(I)) denote the family of boxes of dimension with intevals fom I. The family J 1 consists of a single box, J 1 := B(I 1 ). Fo t > 1, let J t be the set of boxes fom B(I t ) that ae not coveed by any membe of B(I t 1 ). Since B(I g ) = ( ) [n], the boxes J1 J q cove the whole hypegaph. By definition, J 1 = 1. Fo t > 1 we can give a (geneous) uppe bound fo the size of J t as follows: The intevals fom I t defining a membe of J t cannot be spead out into intevals of I t 1. So fist, select two subintevals of a membe of I t 1 and then abitaily othe ( 2) membes of I t. One can do this in at most ( )( ) t J t I t 1 < t 2 t( 2) = t( 1) diffeent ways. The weight of each box fom J t is bounded above by A(n/ t ) c. Hence w(jt ) 1 t g (An c ) t( 1 c).

6 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Poof of Theoem 2.2 Thoughout this section, H is a convex geometic n-vetex -gaph, with cyclic odeing < on the vetices. A subgaph G of H is a split subgaph if thee exists a patition of V (G) into cyclic intevals X 1, X 2,..., X 1 such that fo some i [ 1], evey edge e of G has two vetices in X i and one vetex in evey X j : j i. Let v(h) = e H e and d(h) = e(h)/v(h) 1 denote the codegee density of H. Ou goal is to pove the following Theoem. Theoem 4.1. Fo evey 3 thee exists c 52 contains a split subgaph G with d(g) c d(h). such that evey convex geometic -gaph H We make no attempt to detemine the optimal value of the constant c in this theoem; it is not had to show that c = e Ω(). It is staightfowad to deive Theoem 2.2 fom this theoem. 4.1 Weighted hypegaphs The poof of Theoem 4.1 is inductive, and fo the induction to wok, we appeal to weighted hypegaphs defined in the pevious section. The -sets of positive weight fom a hypegaph on X which we denote by H(ω), and we let V (ω) be the union of all edges in H(ω) and ω = e H(ω) ω(e). We may think of V (ω) as the vetex set of H(ω), and we let v(ω) = V (ω). When the ange of ω is {0, 1}, then ω = H(ω) is the numbe of edges in H(ω). Futhemoe, fo any -gaph H on X, if ω(e) = 1 if e H and ω(e) = 0 othewise, then H(ω) = H, so any hypegaph can be ealized as a weighted hypegaph. The codegee density of ω is defined by d(ω) = ω. (1) v(ω) 1 If G is a subgaph of H(ω), let ω G be defined by ω G (e) = ω(e) fo e G and ω G (e) = 0 othewise. This is the estiction of ω to G. Note that if ω : X {0, 1}, then the codegee density of ω is exactly the codegee density of H(ω). We obtain Theoem 2.2 fo an odeed -gaph H by defining ω(e) = 1 fo e E(H) and ω(e) = 0 othewise. 4.2 Bipatite subgaphs Let us say a convex geometic hypegaph H is bipatite if thee exists an inteval X such that evey edge of H has exactly one vetex in X. We fist pove a lemma on bipatite subgaphs of convex geometic -gaphs, and then use the lemma to commence a poof of a weighted genealization of Theoem 4.1 by induction on. Lemma 4.2. Let 3 and let ω be a weighted convex geometic -gaph. Then thee exists a bipatite G H(ω) such that d(ω G ) d(ω)/ 5.

7 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 7 Poof. The poof of the lemma is by induction on v(ω). If v(ω) 5, then we can set G to be an -set of maximum weight. To see this, note that ω G (e) ω / ( ) v(ω)! ω /v(ω). Since v(ω G ) v(ω), d(ω G ) = ω G v(ω G ) 1! ω! v(ω) 2 1 v(ω) d(ω). Finally, use the fact that v(ω) 5 to get d(ω G ) d(ω)/ 5, as equied. Suppose v(ω) > 5. Patition V (H(ω)) into intevals X 1, X 2,..., X such that X 1 X 2 X X Let H j be the bipatite subgaph of all edges of H with exactly one vetex in X j and put ω j = ω Hj. If ω j ω / 5 fo some j [], then d(ω j ) d(ω)/ 5, and G = H j is the equied bipatite subgaph. If ω j < ω / 5 fo all j [], let F = H(ω)\ j=1 H j. Then ω F ω ω j > j=1 ( ) ω = c ω. Fo S [] of size /2, let ω S be the weighted hypegaph defined by ω S (e) = ω(e) if e X j 2 fo evey j S, and ω S (e) = 0 othewise. By the pigeonhole pinciple, ω S > ω F ) > c ) ω ( /2 ( /2 fo some S [] of size /2. Now evey edge in H(ω S ) is disjoint fom evey X j : j S, so v(ω S ) v(ω) j S X j v(ω) j S v(ω) v(ω) +. 2 By induction, thee is a bipatite G H(ω S ) such that d(ω G ) d(ω S )/ 5. Finally, d(ω S ) = ω S v(ω S ) 1 ( /2 c (2n) ) 1 (v(ω) + ) 1 d(ω). It suffices to show that this is at least d(ω), and so d(ω G ) d(ω)/ 5. To see this, let n = v(ω), note that (1 + /n) 1 e 1/3 e 1/27 when n > 5, and theefoe ( ) ( ) (n + ) 1 e 1/27 n 1. /2 /2 Next note ( /2 ) 2 2/ π < 2 2/3, and theefoe ( ) 8 (n + ) 1 /2 3 e1/27 (2n) 1 < 0.99(2n) 1. Now c c 3 > > 0.99 and so the poposition is poved.

8 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Poof of Theoem 4.1 Using Lemma 4.2 and induction on, we pove the following genealization of Theoem 4.1. Set ψ() = 52. Theoem 4.3. Let 3 and let ω be a weighted convex geometic -gaph. Then thee exist a split -gaph G H(ω) such that d(ω G ) d(ω)/ψ(). Poof. Poceed by induction on. Fo = 3, Lemma 4.2 give the theoem, since in that case a bipatite subgaph is a split subgaph. Fo > 3, pass to a bipatite F H(ω) with d(ω F ) d(ω)/ 5 via Lemma 4.2. Let X and Y be the pats of F, whee evey edge of F intesects X in exactly one vetex. Let F 1 = {e\{x} : e F, x X} and define the new weight function τ by τ(f) = 0 if f F 1 and fo f F 1, τ(f) = e F f e ω(e). We note that F 1 = F 1 (τ) and τ = ω F. Since v(τ) v(ω F ), d(τ) = τ v(τ) 2 v(ω F ) ω F v(ω F ) 1 = v(ω F )d(ω F ). Note that F 1 is ( 1)-unifom, which accounts fo the appeaance of the exta facto v(ω F ). Using d(ω F ) d(ω)/ 5, we find d(τ) v(ω F ) d(ω) 5. By induction, thee exists an almost -patite subgaph E F 1 such that d(τ E ) d(τ) ψ( 1) v(ω d(ω) F ) 5 ψ( 1). (2) Let Z 2, Z 3,..., Z 1 be the pats of E, and let Z 1 = X if v(τ E ) X, othewise let Z 1 be a unifomly selected subset of X of size v(τ E ). Now we define the subgaph we want: let G = {e {z} : e E, z Z 1 }. We claim that with positive pobability, G is the equied almost -patite subgaph, with pats Z 1, Z 2,..., Z 1. We fist pove the following technical poposition: Poposition 4.1. Let m = min{v(τ E ), X }. Then m v(τ E ) 2 v(ω F ) X v(ω G ) 1 2 ( 1). (3) To see this, if m = v(τ E ), then v(ω F ) X and v(ω G ) = 2m, so m v(τ E ) 2 v(ω F ) m 1 X X (v(ω G )/2) 1,

9 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY 12 9 as equied fo (3). Othewise, m = X and v(ω F ) v(ω G ) and v(τ E ) v(ω G )/2, so m v(τ E ) 2 v(ω F ) X v(τ E ) 2 v(ω G ) X (v(ω G )/2) 1 which poves the poposition. By lineaity of expectation, Fix an instance of G with ω G E( ω G ). Then by (2) and (4), E( ω G ) = m τ E. (4) X d(ω G ) = ω G v(ω G ) 1 m τ E X v(ω G ) 1 = m v(τ E) 2 X v(ω G ) 1 d(τ E) m v(τ E) 2 v(ω F ) d(ω) X v(ω G ) 2 5 ψ( 1). Using (3), we obtain d(ω G ) d(ω) ψ( 1). To complete the poof of Theoem 4.3, it suffices to pove that ψ( 1) ψ(). This follows fom ψ( 1) ψ() since 3, Now Theoem 4.1 follows fom Theoem 4.3 by setting ω(e) = 1 fo all e H and ω(e) = 0 othewise, in which case d(h) = d(ω). 5 Poof of Theoem Uppe bound fo k + 1 We stat with the following ecuence: Poposition 5.1. Let 2 k + 1 and n + k. Then ( ) n 2 ex (n, CPk ) + ex (n 2, CP 1 2 k 1 ) + ex (n 1, CPk ). (5) Poof. Let G be an n-vetex odeed -gaph not containing CP k with e(g) = ex (n, CP k ). We may assume V (G) = [n] with the natual odeing. Let G 1 = {e G : {1, 2} e} and G 2 = {e G : 1 e, 2 / e, e {1} {2} G}. Let G 3 be obtained fom G E(G 1 ) E(G 2 ) by gluing vetex 1 with vetex 2 into a new vetex 2.

10 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Since we have deleted the edges of G 1, ou G 3 is an -gaph, and since we have deleted the edges of G 2, G 3 has no multiple edges. Thus e(g) = e(g 1 ) + e(g 2 ) + e(g 3 ). We view G 3 as an odeed -gaph with vetex set {2, 3,..., n}. If G 3 contains a cossing odeed path P with edges e 1, e 2,..., e k, then only e 1 may contain 2, and all othe edges ae edges of G. Thus eithe P itself is in G o the path obtained fom P by eplacing e 1 with e 1 {2 } + {1} o with e 1 {2 } + {2} is in G, a contadiction. Thus G 3 contains no CPk and hence e(g 3 ) ex (n 1, CPk ). (6) By definition, e(g 1 ) ( n 2 2). We can constuct an odeed ( 1)-gaph H2 with vetex set {3, 4,..., n} fom G 2 by deleting fom each edge vetex 1. If H 2 contains a cossing odeed path P with edges e 1, e 2,..., e e i = e i 1 + {2} fo i = 2,..., k foms a CP k k 1, then the set of edges {e 1,..., e k } whee e 1 = e 1 + {1} and in G, a contadiction. Summaizing, we get ex (n, CPk ) = e(g) = e(g 1) + e(g 2 ) + e(g 3 ) ( ) n 2 + ex (n 2, CP 1 2 k 1 ) + ex (n 1, CPk ), as claimed. We ae now eady to pove the uppe bound in Theoem 2.3 fo k + 1: We ae to show that ex (n, CPk ) ( ) ( n n k+1 ). We use induction on k + n. Since CP 1 is simply an edge, ex (n, CP1 ) = 0 fo any n and, and the theoem holds fo k = 1. Suppose now the uppe bound in the theoem holds fo all (k, n, ) with k + n < k + n and we want to pove it fo (k, n, ). By the pevious paagaph, it is enough to conside the case k 2. Then by Poposition 5.1 and the induction assumption, ( ) [( ) ( )] [( ) ( )] n 2 n 2 n k n 1 n k ex (n, CPk ) + + = = [( ) ( ) ( n 2 n 2 n ( ) ( ) n n k + 1, )] [( n k ) + ( n k 1 )] as equied. This poves the uppe bound in Theoem 2.3 fo k Lowe bound fo k + 1. Fo the lowe bound in Theoem 2.3 fo k + 1, we povide the following constuction. Fo 1 k, let G(n,, k) be the family of -tuples (a 1,..., a ) of positive integes such that (a) 1 a 1 < a 2 <... < a n and (b) thee is 1 i k 1 such that a i+1 = a i + 1. Also, let G(n,, + 1) = G(n,, ) {(a 1,..., a ) : a 1 < a 2 <... < a = n}.

11 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Suppose G(n,, k) has a cossing odeed path with edges e 1,..., e k. Let e 1 = (a 1,..., a ) whee 1 a 1 < a 2 <... < a n. By the definition of a cossing odeed path, fo each 2 j k, e j has the fom e j = (a j,1,..., a j, ) whee a i < a j,i < a i+1 fo 1 i j 1 and a j,i = a i fo j i. (7) By the definition of G(n,, k), eithe thee is 1 i k 1 such that a i+1 = a i + 1 o k = + 1 and a = n. In the fist case, we get a contadiction with (7) fo j = i + 1. In the second case, we get a contadiction with (7) fo j = + 1. In ode to calculate G(n,, k), conside the following pocedue Π(n,, k) of geneating all - tuples of elements of [n] not in G(n,, k): take an -tuple (a 1,..., a ) of positive integes such that 1 a 1 < a 2 <... < a n k + 1 and then incease a j by j 1 if 1 j k and by k 1 if k j. By definition, the numbe of outcomes of this pocedue is ( ) n k+1. Also Π(n,, k) neve geneates a membe of G(n,, k) and geneates each othe -subset of [n] exactly once. 5.3 Uppe bound fo k + 2 In this section we apply Theoem 2.1 to pove the uppe bound in Theoem 2.3 fo k + 2. This follows quickly fom the following poposition: Poposition 5.2. Fo k 1, 2, z (n, CP k ) = O(n 1 ). Poof. We pove a stonge statement by induction on k: if H is an odeed n-vetex -gaph with an inteval -coloing with pats X 1, X 2,..., X of size n 1, n 2,..., n, and H has no cossing k-path, then 1 e(h) k n i. n i i=1 Let f(k) be this uppe bound and let P = i=1 n i. The base case k = 1 is tivial. Fo the induction step, assume the esult holds fo paths of length at most k 1, and suppose e(h) > f(k). Fo each ( 1)-set S of vetices mak the edge S {w} whee w is maximum. Let H be the -gaph of unmaked edges. Since we maked at most f(k)/k edges, e(h ) > f(k 1). By the induction assumption thee exists a CP k 1 = v 1v 2... v k+ 2 H and we can extend this to a CP k in H using the maked edge obtained fom the ( 1)-set {v k,..., v k+ 2 }. This poves the poposition. Poposition 5.2 and Theoem 2.1 give ex (n, CP k ) = O(n 1 log n) fo all k 2 as equied. i=1 5.4 Lowe bound fo k + 2 We now tun to lowe bound in Theoem 2.3. Let G(n,, +2) be the family of -tuples (a 1,..., a ) of positive integes such that (a) (b) 1 a 1 < a 2 <... < a n and a 2 a 1 = 2 p, whee p log 2 (n/4) is an intege.

12 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY The numbe of choices of a 1 n/4 is n/4, then the numbe of choices of a 2 is log 2 (n/4), and the numbe of choices of the emaining ( 2)-tuple (a 3,..., a ) is at least ( n/2 2). Thus if 3 and n > 20, then G(n,, + 2) n 1 ( 2)!3 log 2 n. (8) Suppose G(n,, + 2) contains a CP +2 with vetex set {a 1,..., a 2+1 } and edge set {a i... a i+ 1 : 1 i + 2}. By the definition of odeed path, the vetices ae in the following ode on [n]: a 1 < a +1 < a 2+1 < a 2 < a +2 < a 3 < a +3 <... < a < a 2. (9) Hence the 2nd, + 1st and + 2nd edges ae {a +1, a 2, a 3..., a }, {a +1, a +2..., a 2 }, {a 2+1, a +2,..., a 2 }. The diffeences between the second and the fist coodinates in these thee vectos ae d 1 = a 2 a +1, d 2 = a +2 a +1, d 3 = a +2 a 2+1. By (9), it impossible fo each of the thee diffeences d 1, d 2, d 3 to be powes of two. This yields the lowe bound in Theoem 2.3 fo k Poof of Theoem 2.4 In this section, we fist apply Theoem 2.2 to pove Theoem 2.4 fo k 2 1: we will show ex (n, CP k ) k52 n 1. Since fo k, 2, the extemal function ex(n, Pk ) fo an -unifom tight path is Ω(n 1 ), and ex (n, CPk ) ex(n, P k ), we have ex (n, CPk ) = Θ(n 1 ) fo k 2 1. In the case k = + 1, we have ( ) ( ) n n ex (n, CPk ) ex (n, CPk ) =. On the othe hand, ex (n, CP k ) ex (n, CM 2 ) = ( ) n ( n so the second statement in Theoem 2.4 follows. Fo k 2, we have ex (n, CP k ) ex (n, CP k ) = O(n 1 log n) fom Theoem 2.3; so to pove Theoem 2.4 fo k 2, we only need a matching constuction. ), 6.1 Uppe bound fo k 2 1 Given a convex geometic -gaph H with e(h) > k 52 n 1, we apply Theoem 2.2 to obtain a split subgaph G H whee e(g) > kv(g) 1. Let X 0 < X 1 < < X 3 < X be cyclic intevals

13 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY such that evey edge of G contains two vetices in X and one vetex in each X i : 0 i 3. Ou main poposition is as follows: Poposition 6.1. Fo k [2 1], G contains a cossing k-path v 0 v 1... v k+ 2 such that v i X i fo i 1, 2 mod and v i X fo i 1, 2 mod. Poof. We poceed by induction on k, whee the base case k = 1 is tivial. Fo the induction step, suppose that 1 k 2 2, and we have poved the esult fo k and we wish to pove it fo k + 1. Suppose that k i 0, 1 (mod ) whee i <. Fo each f G that has no vetex in X i 1, delete the edge f v G whee v is the lagest vetex in X i 1 in clockwise ode. Let G be the subgaph that emains afte deleting these edges. Then e(g ) e(g) m 1 > (k +1)m 1 m 1 = km 1, so by induction thee is a Pk in G with vetices v 0, v 1,..., v k 1,..., v k+ 2, whee v i X i fo i 1, 2 (mod ) and v i X fo i 1, 2 (mod ). Let v = v k+ 1 be the vetex in X i 1 fo which the edge e k = v k v k+1... v k+ 1 was deleted in foming G. Note that v exists as v k 1 v k... v k+ 2 E(G) and so v k... v k+ 2 G. Adding vetex v and edge e k to ou copy of Pk yields a copy of P k+1 as equied. Next suppose that i 0, 1 (mod ). In fact, we may assume that k { 1, }, and we ae tying to add vetex v = v k+ 1 {v 2 2, v 2 1 } as above but now we want v X. Suppose that k = 1 and we ae tying to add the vetex v = v 2 2. Poceed exactly as befoe, except that when we have f G that has exactly one vetex in X, we choose v to be the lagest vetex in X such that v < v 1 and f {v} G. Such a v cetainly exists due to the edge e = f {v 2 } G. Fo the case k =, we choose v to be the lagest vetex in X which again exists. 6.2 Lowe bound fo k 2 We take the same family G(n,, + 2) as used fo odeed hypegaphs (see Section 5.4), but in the cyclic odeing of the vetex set. When we have a k-edge cossing path P = w 1 w 2... w +k 1, the vetex w 1 does not need to be the leftmost in the fist edge w 1... w, so the agument above does not go though fo k = + 2. In fact, G(n,, + 2) does contain CPk fo k 2 1. Howeve, it does not have a cossing ( + 1)-edge path in which the fist vetex is the second left in the fist edge (epeating the agument in Subsection 5.4). This implies that G(n,, + 2) does not contain CP2 : If it has such a path P = w 1... w 3 1 and w 1 is the ith smallest in the fist edge, then w 2 is the (i + 1)st smallest in the second edge and so on (modulo ). Thus, fo some 1 j, vetex w j is the second left in the jth edge, and the subpath of P stating fom the jth edge has at least + 1 edges. 7 Concluding emaks A hypegaph F is a foest if thee is an odeing of the edges e 1, e 2,..., e t of F such that fo all i {2, 3,..., t}, thee exists h < i such that e i j<i e j e h. It is not had to show that ex(n, F ) = O(n 1 ) fo each -unifom foest F. It is theefoe natual to extend the Pach-Tados Conjectue A to -gaphs as follows:

14 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Conjectue B. Let 2. Then fo any odeed -unifom foest F with inteval chomatic numbe, ex (n, F ) = O(n 1 polylog n). Theoem 2.2 shows that to pove Conjectue B, it is enough to conside the setting of -gaphs of inteval chomatic numbe. Theoem 2.3 veifies this conjectue fo cossing paths, and also shows that the log n facto in Theoem 2.2 is necessay. It would be inteesting to find othe geneal classes of odeed -unifom foests fo 3 fo which Conjectue B can be poved. A elated poblem is to detemine fo which odeed foests F we have ex (n, F ) = O(n 1 )? This is a hypegaph genealization of Baß question [2]. Theoem 2.2 can be used to pove this uppe bound fo many odeed foests othe than just CPk. It appeas to be substantially moe difficult to detemine the exact value of the extemal function fo -unifom cossing k-paths in the convex geometic setting than in the odeed setting. It is possible to show that fo k 2 1, c(k, ) = lim n ex (n, CP ) k ) exists. We have poved seveal nontivial uppe and lowe bounds fo c(k, ) that will be pesented in fothcoming wok, howeve, we do not as yet know the value of c(k, ) fo any pai (k, ) with 2 k, even though in the odeed setting Theoem 2.3 captues the exact value of the extemal function fo all k + 1, and c( + 1, ) =. Let CM k denote the convex geometic gaph consisting of k paiwise cossing line segments. Capoyleas and Pach [5] poved the following theoem which extended a esult of Ruzsa (he poved the case k = 3) and settled a question of Gätne and conjectue of Peles [16]: ( n 1 Theoem 7.1 (Capoyleas-Pach [5]). Fo all n 2k 1, ex (n, CM k ) = 2(k 1)n ( 2k 1 2 Fo 2, an -unifom cossing k-matching CMk is an odeed -gaph whose vetex set is v 0, v 1,..., v k 1, edges ae {v i, v i+k,..., v i+( 1)k } fo 0 i k 1, and vetex odeing v 0 < v 1 < < v k 1. The same definition woks in the convex geometic setting with < a cicula odeing of the vetices. Thus CM2 is pecisely the cossing matching CM. It is not had to see that ex (n, CM k ) = ex (n, CM k ) fo all n, k, 2. We can pove that, unlike the esults on the paths in Theoem 2.3, thee ae no exta log n factos in the fomulas fo cossing matchings and we have ex (n, CM k ) = Θ(n 1 ). We will pesent shape bounds in fothcoming wok. ). Acknowledgement. This eseach was patly conducted duing AIM SQuaRes (Stuctued Quatet Reseach Ensembles) wokshops, and we gatefully acknowledge the suppot of AIM. Refeences [1] B. Aonov, V. Dujmovič, P. Moin, A. Ooms, L. da Silveia, Moe Tuán-type theoems fo tiangles in convex point sets,

15 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY [2] P. Baß, Tuán-type extemal poblems fo convex geometic hypegaphs. Contempoay Mathematics, 342, 25 34, [3] P. Baß, G. Káolyi, P. Valt, A Tuán-type extemal theoy of convex geometic gaphs, Goodman-Pollack Festschift, Spinge 2003, [4] P. Baß, G. Rote, K. Swanepoel, Tiangles of extemal aea o peimete in a finite plana point set. Discete Comp. Geom., 26 (1), 51 58, [5] V. Capoyleas, J. Pach, A Tuán-type theoem fo chods of a convex polygon, J. Combin. Theoy Se. B, 56, [6] P. Edős, On Sets of Distances of n Points, Ame. Math. Monthly 53 (1946), pp [7] Z. Füedi, The maximum numbe of unit distances in a convex n-gon, J. Combin. Theoy Se. A, 55 (1990), [8] Z. Füedi, T. Jiang, A. Kostochka, D. Mubayi, J. Vestaete, Tight paths in convex geometic hypegaphs, [9] W. T. Gowes, E. Long, The length of an s-inceasing sequence of -tuples. axiv: , [10] H. Hopf and E. Pannwitz: Aufgabe N. 167, Jahesbeicht d. Deutsch. Math. Veein. 43 (1934), 114. [11] D. Koándi, G. Tados, I. Tomon, C. Weidet, [12] Y. S. Kupitz, M. Peles, Extemal theoy fo convex matchings in convex geometic gaphs, Discete Comput Geom. 15, (1996), [13] A. Macus, G. Tados, Excluded pemutation matices and the Stanley-Wilf conjectue, Jounal of Combinatoial Theoy, Se. A 107 (2004), [14] J. Pach, G. Tados, Fobidden paths and cycles in odeed gaphs and matices, Isael Jounal of Mathematics 155 (2006), [15] J. Pach, R. Pinchasi, How many unit equilateal tiangles can be geneated by n points in geneal position? Ame. Math. Monthly 110 (2003), [16] M. Peles, unpublished. [17] J. W. Sutheland, Lösung de Aufgabe 167, Jahesbeicht Deutsch. Math.-Veein. 45 (1935), [18] G. Tados, Extemal theoy of odeed gaphs, Poceedings of the Intenational Congess of Mathematics 2018, Vol. 3,

16 Füedi, Jiang, Kostochka, Mubayi, and Vestaëte: Cossing paths, JULY Zoltán Füedi Alféd Rényi Institute of Mathematics Hungaian Academy of Sciences Reáltanoda utca H-1053, Budapest, Hungay Alexand Kostochka Univesity of Illinois at Ubana Champaign Ubana, IL and Sobolev Institute of Mathematics Novosibisk , Russia. Tao Jiang Depatment of Mathematics Miami Univesity Oxfod, OH 45056, USA. Dhuv Mubayi Depatment of Mathematics, Statistics and Compute Science Univesity of Illinois at Chicago Chicago, IL Jacques Vestaëte Depatment of Mathematics Univesity of Califonia at San Diego 9500 Gilman Dive, La Jolla, Califonia , USA.