On ordered Ramsey numbers of tripartite 3-uniform hypergraphs ∗

On ordered Ramsey numbers of tripartite 3-uniform hypergraphs

Martin Balko^{1 †} Máté Vizer^{2 3 ‡} September 27, 2021

1Department of Applied Mathematics,

Faculty of Mathematics and Physics, Charles University, Czech Republic balko@kam.mff.cuni.cz

2 Alfréd Rényi Institute of Mathematics, ELKH, Budapest, Hungary

3 Department of Computer Science and Information Theory, Budapest University of Technology and Economic

vizermate@gmail.com

Abstract

For an integerk ≥ 2, an ordered k-uniform hypergraph H = (H, <) is a k-uniform hypergraphH together with a fixed linear ordering<of its vertex set. Theordered Ramsey number R(H,G) of two orderedk-uniform hypergraphs HandG is the smallestN ∈N such that every red-blue coloring of the hyperedges of the ordered complete k-uniform hypergraph K^(k)_N onN vertices contains a blue copy of Hor a red copy ofG.

The ordered Ramsey numbers are quite extensively studied for ordered graphs, but little is known about ordered hypergraphs of higher uniformity. We provide some of the first nontrivial estimates on ordered Ramsey numbers of ordered 3-uniform hypergraphs.

In particular, we prove that for alld, n∈Nand for every ordered 3-uniform hypergraph Honnvertices with maximum degreedand with interval chromatic number 3 there is an ε=ε(d)>0 such that

R(H,H)≤2^O(n2−ε).

In fact, we prove this upper bound for the numberR(G,K⁽³⁾₃ (n)), whereGis an ordered 3-uniform hypergraph withnvertices and maximum degreedandK₃⁽³⁾(n) is the ordered complete tripartite hypergraph with consecutive color classes of size n. We show that this bound is not far from the truth by provingR(H,K₃⁽³⁾(n))≥2^Ω(nlogn)for some fixed ordered 3-uniform hypergraphH.

∗An extended abstract version of this paper appeared in the Proceedings of Eurocomb 2021, see [4].

†The first author was supported by the grant no. 18-13685Y of the Czech Science Foundation (GAČR) and by the Center for Foundations of Modern Computer Science (Charles University project UNCE/SCI/004). This article is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810115).

‡The second author was supported by the Hungarian National Research, Development and Innovation Office – NKFIH under the grant SNN 129364, KH 130371 and FK 132060, by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences and by the New National Excellence Program under the grant number ÚNKP-21-5-BME-361.

1 Introduction

For an integerk≥2 and ak-uniform hypergraphH, theRamsey number R(H) is the minimum N ∈Nsuch that every 2-coloring of the hyperedges of the completek-uniform hypergraphK_N^(k) onN vertices contains a monochromatic subhypergraph isomorphic toH. Estimating Ramsey numbers is a notoriously difficult problem. Despite many efforts in the last 70 years, no tight bounds are known even for the complete graphK_n onnvertices. Apart from some smaller term improvements, essentially the best known bounds are 2^n/2≤R(K_n)≤2²ⁿ by Erdős [17]

and by Erdős and Szekeres [14]. The Ramsey numbersR(Kn^(k)) are even less understood for k≥3. For example, it is only known that

2^Ω(n2)≤R(K_n⁽³⁾)≤2^2O(n), (1) as shown by Erdős, Hajnal, and Rado [16]. A famous conjecture of Erdős, for whose proof Erdős offered $500 reward, states that there is a constantc >0 such thatR(Kn⁽³⁾)≥2^2cn.

Recently, a variant of Ramsey numbers for hypergraphs with a fixed order on their vertex sets has been introduced [2, 6]. For an integerk≥2, an ordered k-uniform hypergraph H is a pair (H, <) consisting of a k-uniform hypergraphH and a linear ordering <of its vertex set. An orderedk-uniform hypergraphH₁= (H1, <1) is an ordered subhypergraphof another orderedk-uniform hypergraphH₂ = (H₂, <₂), written H₁ ⊆ H₂, ifH₁ is a subhypergraph of H2 and<1 is a suborder of <2. Two ordered hypergraphs H₁ andH₂ areisomorphic if there is an isomorphism between their underlying hypergraphs that preserves the vertex orderings ofH₁ andH₂. Note that, up to isomorphism, there is a unique ordered completek-uniform hypergraphK^(k)n on nvertices.

The ordered Ramsey number R(H,G) of two ordered k-uniform hypergraphs H and G is the smallest N ∈Nsuch that every coloring of the hyperedges of K^(k)_N by colors red and blue contains a blue ordered subhypergraph isomorphic toHor a red ordered subhypergraph isomorphic toG. In the diagonal caseH=G, we just writeR(G) instead ofR(G,G).

The ordered Ramsey numbers are known to be finite and it is easy to see that they grow at least as fast as the standard Ramsey numbers. Studying ordered Ramsey numbers has attracted a lot of attention lately (see the survey by Conlon, Fox, and Sudakov [11]), as there are various motivations coming from the field of discrete geometry. It is known that ordered Ramsey numbers can behave quite differently than the standard Ramsey numbers, especially for sparse ordered graphs [2, 3, 6]. However, so far, the ordered Ramsey numbers have been studied mostly for ordered graphs only and very little is known about ordered Ramsey numbers of orderedk-uniform hypergraphs with k≥3.

In this paper, we focus on 3-uniform hypergraphs and we prove some new bounds on the ordered Ramsey numbers of ordered tripartite 3-uniform hypergraphs. We also pose several new open problems in Section 6.

1.1 Preliminaries

For an orderedk-uniform hypergraph H= (H, <) and two subsets U and V of vertices ofH, we say that U and V are consecutive if all vertices from U precede all vertices of V in <.

An interval in H is a subset I of vertices of H such that for all vertices u, v, w of H with u < v < w andu, w ∈I we have v∈I.

For integersk≥2 and χ≥k, we useKχ^(k)(n) to denote thecomplete k-uniform χ-partite hypergraph, that is, the vertex set of Kχ^(k)(n) is partitioned into χ sets of size n and every k-tuple with at most one vertex in each of these parts forms a hyperedge. The ordering ofKχ^(k)(n), in which the color classes form consecutive intervals, is denoted by Kχ^(k)(n). We sometimes useK_n,n to denoteK⁽²⁾₂ (n).

Thedegreeof a vertexvin a hypergraphHis the number of hyperedges ofHthat containv.

For d∈ N, a k-uniform hypergraphH is d-degenerate if there is an ordering v₁ ≺ · · · ≺v_t of vertices ofH such that each vi is contained in at mostdhyperedges of H that contain a vertex fromv₁, . . . , vi−1.

For a positive integern, we use [n] to denote the set{1, . . . , n}. We omit floor and ceiling sign whenever they are not crucial and we use log and ln to denote base 2 logarithm and the natural logarithm, respectively.

1.2 Previous results

The ordered Ramsey numbers of k-uniform ordered hypergraphs with k ≥3 remain quite unexplored. Only the ordered Ramsey numbers of so-called monotone hyperpaths are well understood due to their close connections to the famous Erdős–Szekeres Theorem [17]; see [1, 19, 22]. Amonotone hyperpath Pn^(k) on nvertices is an ordered k-uniform hypergraph where the hyperedges are formed byk-tuples of consecutive vertices. Note that the maximum degree of ak-uniform monotone hyperpath is at most k. Moshkovitz and Shapira [22] showed that R(P_n^(k)) = tow_k−1((2−o(1))n) fork≥3, where tow_h is thetower function of height hdefined as tow₁(x) =xand tow_h(x) = 2^towh−1(x) forh≥2.

Thus even for 3-uniform hypergraphsHwith bounded maximum degree the numbersR(H) can grow very fast. We get an exponential lower bound onR(H) even for 3-uniform ordered hypergraphsH with maximum degree 3. A similar result is known for ordered graphs, as for arbitrarily large values ofnthere are ordered graphsM_n withnvertices and maximum degree 1 such thatR(M_n)≥n^{Ω(logn/log logn)} [2, 6]. This superpolynomial growth rate is in sharp contrast with the situation for unordered hypergraphs, where the Ramsey numberR(H) of everyk-uniform hypergraph H with bounded kand with bounded maximum degree is at most linear in the number of vertices ofH [5, 7, 12, 13, 20, 23].

Therefore, in order to obtain smaller upper bounds on the ordered Ramsey numbers, it is necessary to bound other parameter besides the maximum degree. A natural choice is so-called interval chromatic number, which can be understood as an analogue of the chromatic number for ordered graphs due to a variant of the Erdős–Stone–Simonovits theorem for ordered graphs proved by Pach and Tardos [25]. Theinterval chromatic number χ<(H) of an ordered k-uniform hypergraph H is the minimum number of intervals the vertex set of H can be partitioned into so that each hyperedge ofHhas at most one vertex in each of the intervals.

For ordered graphs, bounding both parameters indeed helps, as the ordered Ramsey numberR(G) of every ordered graphGwith bounded maximum degreedand bounded interval chromatic numberχ is at most polynomial in the number of vertices [2, 6]. SinceG ⊆ K_χ⁽²⁾(n), this result follows from the following stronger estimate proved by Conlon, Fox, Lee, and Sudakov [6]: for alld, χ∈N, every d-degenerate ordered graph G on nvertices with interval chromatic numberχ satisfies

R(G,K⁽²⁾_χ (n))≤n^32dlogχ. (2) A natural question is whether we can also get some good upper bounds on ordered Ramsey

numbers of similarly restricted classes of ordered hypergraphs. If the interval chromatic number is bounded, then we can use a result of Conlon, Fox, and Sudakov [9], who showed that, for all positive integersχ≥3 and n,

R(K_χ⁽³⁾(n))≤2^22Rn2,

whereR =R(Kχ−1). Since every ordering of Kχ⁽³⁾(χn) contains an ordered subhypergraph isomorphic toK⁽³⁾_χ (n) and every ordered 3-uniform hypergraph on n vertices with interval chromatic numberχ is an ordered subhypergraph ofK⁽³⁾_χ (n), we obtain the following bound.

Corollary 1. For all positive integers χ≥3 and n, every ordered3-uniform hypergraph H on nvertices with interval chromatic number χ satisfies

R(H)≤2^22Rχ2n2,

where R=R(Kχ−1). In particular, if the interval chromatic number χ of His fixed, we have R(H)≤2^O(n2).

Note that the last bound is asymptotically tight for dense ordered hypergraphs, as a standard probabilistic argument shows that R(H) ≥ 2^Ω(n2) for every ordered 3-uniform hypergraphHonnvertices with Ω(n³) hyperedges. In particular, we getR(K₃⁽³⁾(n))≥2^Ω(n2).

2 Our results

Since the bounds on the ordered Ramsey numbers from Corollary 1 are asymptotically tight for dense ordered hypergraphs with bounded interval chromatic number, we consider the sparse case with bounded maximum degree and interval chromatic number. The situation for ordered hypergraphs seems to be more difficult than for ordered graphs, so we focus on the first nontrivial case, which is for ordered 3-uniform hypergraphs with interval chromatic number 3.

Assuming the maximum degree of an ordered hypergraphHwith χ<(H) = 3 is sufficiently small, we obtain a better upper bound onR(H) than the estimate 2^O(n2) we would get from Corollary 1. We can prove an estimate with a subquadratic exponent even in the more general settingR(H,K⁽³⁾₃ (n)), where, additionally, the interval chromatic number of His arbitrary.

Theorem 2. Let Hbe an ordered 3-uniform hypergraph on t vertices with maximum degreed and let s be a positive integer. Then there are constants C=C(d) and c >0 such that

R(H,K₃⁽³⁾(s))≤t·2^C(s2−1/(1+cd2)). In particular, fors=t=n and bounded d, we get the estimate

R(H,K₃⁽³⁾(n))≤2^O(n2−1/(1+cd2)). (3)

The main idea of the proof of Theorem 2 is based on an embedding lemma from [10], where the authors study Erdős–Hajnal-type theorems for 3-uniform tripartite hypergraphs.

We prove a variant of this lemma, which works for ordered hypergraphs, does not consider

induced copies, and uses the assumption that the maximum degree ofH is bounded instead of assuming that the number of vertices ofH is fixed.

Since every ordered 3-uniform hypergraphH on nvertices withχ_<(H) = 3 is an ordered subhypergraph ofK⁽³⁾₃ (n), we immediately obtain the following corollary.

Corollary 3. Let H be an ordered 3-uniform hypergraph on nvertices with maximum degree dand with interval chromatic number3. Then there exists an ε=ε(d)>0 such that

R(H)≤2^O(n2−ε).

It might seem wasteful to use Theorem 2 in order to obtain Corollary 3, as the ordered hypergraphHis much sparser than K⁽³⁾₃ (n). However, as noted in [6], this intuition is wrong already for ordered graphs, as there are ordered matchingsMon nvertices with χ<(M) = 2 and ordered graphs G on N = 2^nc vertices for some constant c > 0 such that G has edge density at least 1−n^−c and does not containM as an ordered subgraph. In fact, the best known upper bounds onR(G) forn-vertex ordered graphs G with bounded maximum degree andχ_<(G) =χ are derived from the bound (2) onR(G,K⁽²⁾_χ (n)).

The upper bound (3) is quite close to the truth, as even when H is fixed we get a superexponential lower bound onR(H,K⁽³⁾₃ (n)). We note that since the first version of this preprint we learned that independently Fox and He (Theorem 1.3 in [18]) proved the same lower bound for the unordered Ramsey number and that implies the result of Theorem 4.

However we leave this result here as our proof is much simpler.

Theorem 4. For every t≥3 and every positive integer n, we have R(K⁽³⁾_t+1,K₃⁽³⁾(n))≥2^Ω(nlogn).

We do not have any nontrivial lower bound in the diagonal caseR(H) forH with bounded maximum degree and χ_<(H) = 3. We note that even for ordered graphs G with bounded maximum degreedand χ<(G) = 2 the best known lower bound onR(G) (and also onR(H)) is only of order Ω((n/logn)²) [3], while the upper bound onR(G) is of order n^O(d) [2, 6].

Concerning k-uniform hypergraphs withk >3, the following result is based on a modifica- tion of the proof from [8, Proposition 6.3] and gives an estimate on ordered Ramsey numbers of orderedk-uniform hypergraphs with bounded interval chromatic number. In particular, this estimate shows that we do not have a tower-type growth rate forR(H) once the uniformity and the interval chromatic number ofHare bounded.

Proposition 5. Let χ, k be integers with χ ≥ k ≥ 2 and let H be an ordered k-uniform hypergraph on n vertices with interval chromatic number χ. Then there is a constantc such that

R(H)≤2^{Rχ(χ−1)(cχn)χ−1},

where R=R(Kχ^(k)). In particular, if the uniformity k and the interval chromatic number χ of H are fixed, we have

R(H)≤2^O(nχ−1).

Our understanding of the ordered Ramsey numbers of ordered hypergraphs is still very limited. Many interesting open problem arose during our study and we would like to draw attention to some of them in Section 6.

3 Proof of Theorem 2

Here we prove Theorem 2 by giving an upper bound on the ordered Ramsey numbers of ordered 3-uniform hypergraphs with bounded maximum degree versusK₃⁽³⁾(n). We prove the following slightly stronger result.

Theorem 6. Let H be an ordered 3-uniform hypergraph ont vertices with maximum degreed, let s be a positive integer andρ∈(0,1/8) be a real number. Then there is a constant C⁰ such that

R(H,K₃⁽³⁾(s))≤t·2^{C0(s3/2·ρ−30d}

2·d⁶+s²log (_1−4ρ¹ ))

. First, we show that Theorem 6 implies Theorem 2.

Proof of Theorem 2. We can assume that s > 8^2+60d2 by choosing the constant C = C(d) from the statement of Theorem 2 sufficiently large. We then chooseρ=s^{−1/(2+60d2)}, which givesρ <1/8 by our assumption on s. Since 1−z≥e^−2z for everyz with 0≤z≤1/2, we obtain (1−4ρ)^−s2 ≤e^8ρs2 and thus s²log (_1−4ρ¹ ))≤8s²ρ. By our choice of ρ, we then get

s^3/2ρ^−30d2 =s²ρ=s^{2−1/(2+60d2)}. Therefore we can rewrite the upper bound from Theorem 6 as

R(H,K⁽³⁾₃ (s))≤t·2^C0(d6+8)s2ρ≤2^Cs2−1/(2+60d2)

for sufficiently largeC=C(d), which concludes the proof.

In the rest of the section, we prove Theorem 6, but we first state some definitions. For a graphGand two disjoint subsetsX andY of vertices ofG, we use d(X, Y) to denote theedge density between X and Y, that is,d(X, Y) = ^e(X,Y_|X||Y|⁾, where e(X, Y) denotes the number of

edges with one vertex inX and with the other one in Y.

For positive real numbers ε1, ε2, and ρ, we say that an ordered bipartite graph G with consecutive color classesU andV isbi-(ε1, ε2, ρ)-dense betweenU and V, if for all setsX ⊆U and Y ⊆V with|X| ≥ε₁|U|and |Y| ≥ε₂|V|we have d(X, Y)≥ρ.

For positive real numbersεandρand a positive integerm, an ordered 3-uniform hypergraph Histri-(ε, ρ, m)-dense, if for all consecutive subsets V1, V2, V3 of vertices ofH, each of size at mostm, and for all bipartite graphs G_1,2, G_1,3, G_2,3, eachG_i,j betweenV_i andV_j, for which there are at least εm³ triangles with one edge in each Gi,j, at least ρ-proportion of these triangles forms hyperedges inH.

The following embedding lemma is based on a similar result from [10].

Lemma 7. Let H be an ordered 3-uniform hypergraph ont vertices with maximum degree d.

Letε >0 andρ∈(0,1) be two real numbers with ε≤2⁻⁶·ρ^15d2 ·d⁻³. IfG is a tri-(ε, ρ, n/t)- dense ordered 3-uniform hypergraph on n≥t/ε vertices, thenH is an ordered subhypergraph of G.

Proof. Letv₁≺ · · · ≺v_tbe the vertices ofH= (H,≺). For a vertexv_j ofHwithj > i, we use degⁱ_H(j) to denote the number^P_{e∈E(H):vj∈e}|e∩ {v₁, . . . , v_i}|. Similarly, for verticesv_j andv_k ofH andi < j, k, we use degⁱ_H(j, k) to denote the number of hyperedges of H containingv_j, v_k, and a vertex from {v₁, . . . , v_i}. Observe that degⁱ_H(j)≤2dand degⁱ_H(j, k)≤dfor allj

andk withi < j, k, as the maximum degree of His dand each hyperedge is multiplied by at most 2.

We partition the vertex set ofG into consecutive intervals U₁, . . . , U_t, each of size n/t. We embed a copyf(H) ofHin G one vertex at time, embedding each vertex f(v_i) off(H) toU_i. We proceed by induction oni= 0,1, . . . , t−1. Assuming that the vertices v1, . . . , vi have been embedded asf(v₁), . . . , f(v_i), we show that there are sets U_i+1ⁱ , . . . , U_tⁱ and graphs Gⁱ_j,k with i < j < k ≤t such that the following three conditions are satisfied:

(i) |U_jⁱ| ≥C_jⁱ·n/t, where C_jⁱ=ρ^d·degiH(j),

(ii) Gⁱ_j,k is a bipartite graph between U_jⁱ and U_kⁱ, which is bi-(εⁱ_j, εⁱ_k, ρⁱ_j,k)-dense between U_jⁱ and U_kⁱ with εⁱ_j = ρ^{4d2−d·degiH(j)}/(4d), εⁱ_k = ρ^{4d2−d·degiH(k)}/(4d), and ρⁱ_j,k =ρ^degiH(j,k), and

(iii) for everyh ≤i, every edge ofGⁱ_j,k forms a hyperedge of G with f(v_h) if{v_h, v_j, v_k} is a hyperedge of H. Also, for all h1 and h2 with h1 < h2 ≤i, every vertexu∈U_jⁱ with i < j forms a hyperedge {f(v_h1), f(v_h2), u} ofG if{v_h1, v_h2, v_j} is a hyperedge ofH.

For the induction base, assumei= 0 and setU_j⁰=Uj for every j∈ {1, . . . , t}. For allj andk with 1≤j < k≤t, we letGⁱ_j,k be the complete bipartite graph between U_j⁰ andU_k⁰. Then the three conditions are trivially satisfied.

For the induction step, assume that the vertices v₁, . . . , v_i have been embedded for some i≥0 while maintaining conditions (i), (ii), and (iii). We show how to embed the vertexvi+1.

We let W_i+1 be the set of vertices w fromU_i+1ⁱ such that the neighborhood U_jⁱ(w) ofw in Gⁱ_i+1,j has size at least ρⁱ_i+1,j|U_jⁱ| for every j ∈ {i+ 2, . . . , t} such that v_i+1 and v_j are contained in a hyperedge ofH. Since the graph Gⁱ_i+1,j is bi-(εⁱ_i+1, εⁱ_j, ρⁱ_i+1,j)-dense, there are at mostεⁱ_i+1|U_i+1ⁱ | ≤εⁱ_i+1n/tvertices in U_i+1 that have less than ρⁱ_i+1,j|U_jⁱ|neighbors in |U_jⁱ|.

Using condition (i) and the fact that the maximum degree ofH is d, we see that the size ofWi+1 satisfies

|W_i+1| ≥ |U_i+1ⁱ | −2dεⁱ_i+1|U_i+1ⁱ | ≥ |U_i+1ⁱ | −2dεⁱ_i+1n/t≥C_i+1ⁱ n/t−2dεⁱ_i+1n/t≥C_i+1ⁱ n/(2t), where the last inequality follows from εⁱ_i+1≤C_i+1ⁱ /(4d), as degⁱ_H(i+ 1)≤2d.

For a vertexw∈W_i+1and indicesjandksuch thati+1< j < kand{v_i+1, v_j, v_k} ∈E(H), we define the graphH_j,k(w) as a subgraph of Gⁱ_j,k between U_jⁱ(w) and U_kⁱ(w) consisting of edges {x, y} such that {w, x, y} ∈E(G). We also letW_j,kⁱ⁺¹ be the set of vertices w∈ Wi+1

such that the graphH_j,k(w) is not bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-dense between U_jⁱ(w) and U_kⁱ(w).

By the definition of bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-density, for everyw∈W_j,kⁱ⁺¹, there are setsY_j(w)⊆ U_jⁱ(w) and Yk(w) ⊆ U_kⁱ(w) such that |Y_j(w)| ≥ εⁱ⁺¹_j |U_jⁱ(w)|, |Y_k(w)| ≥ εⁱ⁺¹_k |U_kⁱ(w)|, and d(Yj(w), Y_k(w))< ρⁱ⁺¹_j,k inH_j,k(w). Since w∈Wi+1, we have

|Y_j(w)| ≥εⁱ⁺¹_j |U_jⁱ(w)| ≥εⁱ⁺¹_j ·ρⁱ_i+1,j|U_jⁱ| ≥εⁱ_j|U_jⁱ|,

where the last inequality follows from εⁱ⁺¹_j ·ρⁱ_i+1,j ≥ εⁱ_j, as, since {v_i+1, v_j, v_k} ∈E(H), we have degⁱ⁺¹_H (j)>degⁱ_H(j) and degⁱ_H(i+ 1, j)≤d. Analogously, we obtain|Y_k(w)| ≥εⁱ_k|U_kⁱ|.

We let Ji+1,j be the graph connecting each w ∈ W_j,kⁱ⁺¹ to vertices from Yj(w) and we analogously define the graphJ_i+1,k. The graph Gⁱ_j,k is bi-(εⁱ_j, εⁱ_k, ρⁱ_j,k)-dense between U_jⁱ and

U_kⁱ by condition (ii). Thus, since|Y_j(w)| ≥εⁱ_j|U_jⁱ|and|Y_k(w)| ≥εⁱ_k|U_kⁱ|, the number of triangles in the tripartite graph betweenW_j,kⁱ⁺¹,U_jⁱ, and U_kⁱ formed byJ_i+1,j,J_i+1,k, and Gⁱ_j,k is at least ρⁱ_j,k^P_w∈Wⁱ⁺¹

j,k

|Y_j(w)||Y_k(w)|.

Suppose for contradiction that|W_j,kⁱ⁺¹| ≥ |W_i+1|/(2d) for somej and kwith i+ 1< j < k and {v_i+1, vj, v_k} ∈ E(H). Then, since |Y_j(w)| ≥ εⁱ_j|U_jⁱ| and |Y_k(w)| ≥ εⁱ_k|U_kⁱ| for every w∈W_j,kⁱ⁺¹,

ρⁱ_j,k ^X

w∈W_j,kⁱ⁺¹

|Y_j(w)||Y_k(w)| ≥ρⁱ_j,kεⁱ_jεⁱ_k|W_j,kⁱ⁺¹||U_jⁱ||U_kⁱ| ≥ρⁱ_j,kεⁱ_jεⁱ_kC_jⁱC_kⁱ|W_j,kⁱ⁺¹|(n/t)²,

where the last inequality follows from condition (i). Using the assumption|W_j,kⁱ⁺¹| ≥ |W_i+1|/(2d) and the fact|W_i+1| ≥C_i+1ⁱ n/(2t), we can estimate the above expression from below by

ρⁱ_j,kεⁱ_jεⁱ_kC_jⁱC_kⁱC_i+1ⁱ n³/(4dt³)

Our choice of parameters then givesρⁱ_j,kεⁱ_jεⁱ_kC_jⁱC_kⁱC_i+1ⁱ /(4dt³)≥ε/t³. Altogether, there are at least

ρⁱ_j,k ^X

w∈W_j,kⁱ⁺¹

|Y_j(w)||Y_k(w)| ≥ε(n/t)³

triangles in the tripartite graph betweenW_j,kⁱ⁺¹, U_jⁱ, and U_kⁱ formed by Ji+1,j, J_i+1,k, and Gⁱ_j,k. Thus, since the ordered hypergraph G is tri-(ε, ρ, n/t)-dense, at least ρ-proportion of these triangles forms hyperedges inG. Therefore there are at leastρ·ρⁱ_j,k^P_w∈Wⁱ⁺¹

j,k

|Y_j(w)||Y_k(w)|

hyperedges ofG betweenW_j,kⁱ⁺¹,U_jⁱ, and U_kⁱ.

On the other hand, the number of hyperedges of G containing a vertex w∈ W_j,kⁱ⁺¹ and having an edge in each of the graphsJ_i+1,j, J_i+1,k, andGⁱ_j,k is the number of edges ofH_j,k(w) betweenYj(w) and Yk(w). This number of edges is less thanρⁱ⁺¹_j,k |Y_j(w)||Y_k(w)|, as we know thatd(Y_j(w), Y_k(w))< ρⁱ⁺¹_j,k inH_j,k(w). Thus the number of the hyperedges of G is less than

ρⁱ⁺¹_j,k ^X

w∈W_j,kⁱ⁺¹

|Y_j(w)||Y_k(w)| ≤ρ·ρⁱ_j,k ^X

w∈W_j,kⁱ⁺¹

|Y_j(w)||Y_k(w)|,

where the least inequality follows from ρⁱ⁺¹_j,k ≤ ρ·ρⁱ_j,k, as {v_i+1, v_j, v_k} ∈ E(H) and thus we have degⁱ⁺¹_H (j, k) > degⁱ_H(j, k). This contradicts the fact that there are at least ρ · ρⁱ_j,k^P_w∈Wⁱ⁺¹

j,k

|Y_j(w)||Y_k(w)|such hyperedges of G.

Thus|W_j,kⁱ⁺¹|<|W_i+1|/(2d). In particular, the number of vertices w∈W_i+1 that do not lie in any setW_j,kⁱ⁺¹ such that i+ 1< j < k≤tand {v_i+1, vj, vk} ∈E(H) is at least

|W_i+1| − ^X

j<k:{v_i+1,vj,vk}∈E(H)

|W_j,kⁱ⁺¹|>|W_i+1| − d|W_i+1|

2d = |W_i+1| 2 ,

as we are summing over at most d pairs (j, k), because the maximum degree of H is d.

Since |W_i+1| ≥ C_i+1ⁱ n/(2t), we have at least C_i+1ⁱ n/(4t) such vertices and we let f(vi+1) be any of them. Since n ≥ t/ε > 4t/C_i+1ⁱ , at least one such vertex indeed exists. For every j ∈ {i+ 2, . . . , t} such that v_i+1 and v_j are contained in a hyperedge of H, we let

U_jⁱ⁺¹ =U_jⁱ(f(v_i+1)). We keepU_jⁱ⁺¹=U_jⁱ for all other valuesj. Letj and kbe indices such thati+ 1 < j < k≤t. If {v_i+1, vj, vk} ∈E(H), we set Gⁱ⁺¹_j,k =Hj,k(f(vi+1)). For all other values ofj andk, we letGⁱ⁺¹_j,k be the subgraph ofGⁱ_j,k induced byU_jⁱ⁺¹ andU_kⁱ⁺¹. In particular, if none of the verticesvj andv_k lies in a hyperedge of Hwith vi+1, we have Gⁱ⁺¹_j,k =Gⁱ_j,k.

To finish the induction step, it remains to verify conditions (i), (ii), and (iii). To verify condition (i), first observe that if, forj∈ {i+ 2, . . . , k}, the vertexvj is not in a hyperedge of G with vi+1, then, by the choice of C_jⁱ and C_jⁱ⁺¹, |U_jⁱ⁺¹|= |U_jⁱ| ≥ C_jⁱn/t = C_jⁱ⁺¹n/t, as degⁱ_H(j) = degⁱ⁺¹_H (j). Otherwise |U_jⁱ⁺¹| = |U_jⁱ(f(v_i+1))|. Since f(v_i+1) ∈ W_i+1, we have

|U_jⁱ(f(v_i+1))| ≥ ρⁱ_i+1,j|U_jⁱ|. Since degⁱ_H(j) < degⁱ⁺¹_H (j) and degⁱ_H(i+ 1, j) ≤ d, we have ρⁱ_i+1,jC_jⁱ ≥C_jⁱ⁺¹. So we obtain|U_jⁱ⁺¹| ≥C_jⁱ⁺¹n/t. Thus condition (i) is satisfied.

For condition (ii), let j and kbe indices such that i+ 1< j < k ≤t. Consider first the case when{v_i+1, vj, v_k} ∈ E(H). ThenGⁱ⁺¹_j,k =H_j,k(f(vi+1)). Since f(vi+1) does not lie in any setW_j,kⁱ⁺¹ such that i+ 1< j < k and{v_i+1, v_j, v_k} ∈E(H), the graphH_j,k(f(v_i+1)) is bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-dense betweenU_jⁱ(f(v_i+1)) andU_kⁱ(f(v_i+1)). Thus Gⁱ⁺¹_j,k =H_j,k(f(v_i+1)) is bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-dense betweenU_jⁱ⁺¹ =U_jⁱ(f(vi+1)) andU_kⁱ⁺¹=U_kⁱ(f(vi+1)), which verifies condition (ii) in this case. Now, assume{v_i+1, v_j, v_k}∈/E(H). If none of the two verticesv_jand v_k is in a hyperedge ofHwith v_i+1, then degⁱ⁺¹_H (j) = degⁱ_H(j), degⁱ⁺¹_H (k) = degⁱ_H(k), and degⁱ⁺¹_H (j, k) = degⁱ_H(j, k). In particular,εⁱ_j =εⁱ⁺¹_j , εⁱ_j =εⁱ⁺¹_j , and ρⁱ⁺¹_j,k =ρⁱ_j,k. Since U_jⁱ⁺¹= U_jⁱ,U_kⁱ⁺¹=U_kⁱ, andGⁱ⁺¹_j,k =Gⁱ_j,k is bi-(εⁱ_j, εⁱ_k, ρⁱ_j,k)-dense betweenU_jⁱandU_kⁱ, we see thatGⁱ⁺¹_j,k is bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-dense betweenU_jⁱ⁺¹andU_kⁱ⁺¹, which again verifies condition (ii). It remains to consider the case when exactly one of the verticesvj andvk is in a hyperedge ofHwithvi+1. By symmetry, we can assume without loss of generality thatv_j andv_i+1 are in a hyperedge of H. Then degⁱ⁺¹_H (j) >degⁱ_H(j), degⁱ⁺¹_H (k) = degⁱ_H(k), and degⁱ⁺¹_H (j, k) = degⁱ_H(j, k).

In particular,εⁱ_j > εⁱ⁺¹_j , εⁱ_k =εⁱ⁺¹_k , and ρⁱ⁺¹_j,k =ρⁱ_j,k. By definition, we haveU_jⁱ⁺¹ =U_jⁱ(f(v_i+1)) andU_kⁱ⁺¹ =U_kⁱ. Let X be a subset ofU_jⁱ⁺¹ of size at least εⁱ⁺¹_j |U_jⁱ⁺¹|and let Y be a subset of U_kⁱ⁺¹ of size at least εⁱ⁺¹_k |U_kⁱ⁺¹|=εⁱ_k|U_kⁱ|. We want to show that d(X, Y) ≥ρⁱ⁺¹_j,k inGⁱ⁺¹_j,k . The subset X has size at least εⁱ_j|U_jⁱ|, as|U_jⁱ⁺¹|=|U_jⁱ(f(vi+1))| ≥ ρⁱ_i+1,j|U_jⁱ|and our choice ofεⁱ⁺¹_j together with the fact degⁱ_H(i+ 1, j) ≤dgives εⁱ⁺¹_j |U_jⁱ⁺¹| ≥εⁱ_j|U_jⁱ|. Thus, since Gⁱ_j,k is bi-(εⁱ_j, εⁱ_k, ρⁱ_j,k)-dense betweenU_jⁱ ⊇U_jⁱ⁺¹ andU_kⁱ ⊇U_kⁱ⁺¹, the density between X andY is at leastρⁱ_j,k =ρⁱ⁺¹_j,k inGⁱ_j,k. Since Gⁱ⁺¹_j,k is an induced subgraph of Gⁱ_j,k, the density between X andY is also at least ρⁱ_j,k =ρⁱ⁺¹_j,k inGⁱ⁺¹_j,k . ThusGⁱ⁺¹_j,k is bi-(εⁱ⁺¹_j , εⁱ⁺¹_k , ρⁱ⁺¹_j,k )-dense between U_jⁱ⁺¹ and U_kⁱ⁺¹, which verifies condition (ii).

We show that Condition (iii) is satisfied as well. Let j and k be indices such that i+ 2< j < k ≤t. If {v_h, vj, vk} is a hyperedge of H for some h ≤i+ 1, then {f(v_h), x, y}

is a hyperedge ofG for every edge {x, y} of Gⁱ⁺¹_j,k . This is true for h ≤ i by the induction assumption, asGⁱ⁺¹_j,k ⊆Gⁱ_j,k. For h=i+ 1 we haveGⁱ⁺¹_j,k =Hj,k(f(vi+1)) and the claim holds by the definition of H_j,k(f(vi+1)). Similarly, for all h₁ and h₂ with h₁ < h₂ ≤i+ 1, every vertex u ∈ U_jⁱ⁺¹ with i+ 1 < j forms a hyperedge {f(v_h1), f(v_h2), u} of G if {v_h1, v_h2, v_j} is a hyperedge of H. This is because if h2 ≤ i, then the claim follows from the induction assumption and the factU_jⁱ⁺¹ ⊆U_jⁱ. Forh2=i+ 1, the triple{f(v_h1), f(v_h2), x}is a hyperedge ofG for every x∈U_jⁱ⁺¹ =U_jⁱ(f(v_h2)) by the inductive assumption.

Finally, after we find all verticesf(v1), . . . , f(vt), condition (iii) ensures that they determine a copy ofHas an ordered subhypergraph of G.

We use the following result proved by Conlon, Fox, and Sudakov [10]. It says that if a graphG contains many triangles, a 3-uniform hypergraph whose hyperedges form a dense subset of the set of triangles inG contains a large copy ofK₃⁽³⁾(n).

Lemma 8 ([10]). Let V1, V2, V3 be pairwise disjoint sets of vertices, each of size at most m, and let G_i,j be a bipartite graph between V_i and V_j for all iand j with 1≤i < j ≤3. Assume there are at leastδm³ triangles in the tripartite graph formed by G_1,2, G_1,3, andG_2,3. Let Gbe a 3-uniform hypergraph containing at least (1−η)-proportion of the triangles in the tripartite graph, where 0< η <1/8. Then G contains a copy of K₃⁽³⁾(s) provided that

e^{210δ−2s3/2}(1−4η)^−4s2 δ

16 4s

≤m.

We note that, although it is not explicitly stated in the above lemma, the copy of K₃⁽³⁾(s) hasV_ias theith color class. Thus if the setV₁∪V₂∪V₃ is ordered andV₁, V₂, V₃ are consecutive in this ordering, we actually get a copy of the ordered hypergraphK⁽³⁾₃ (s).

We now proceed with the proof of Theorem 6, which implies Theorem 2.

Proof of Theorem 6. Let H be an ordered 3-uniform hypergraph ontvertices with maximum degree d, let s be a positive integer and ρ ∈ (0,1/8) be a real number. We choose ε = 2⁻⁶·ρ^15d2·d⁻³ and we letN be an integer such that

N ≥t·2^{228·s3/2·ρ−30d}

2·d⁶+12s²log (_1−4ρ¹ )

.

Note thatN ≥t/ε. Consider a red-blue coloringχ of the hyperedges of K⁽³⁾_N . We use G to denote the ordered 3-uniform hypergraph onN vertices formed by the hyperedges ofK⁽³⁾_N that are blue inχ. Similarly, we let G be the hypergraph determined by red hyperedges ofK⁽³⁾_N inχ.

If G is tri-(ε, ρ, N/t)-dense, then, since N ≥ t/ε, Lemma 7 implies that there is a blue copy ofH inχand we are done. Thus we assume that G is not tri-(ε, ρ, N/t)-dense. That is, there are three consecutive subsetsV1, V2, V3 of vertices ofG, each of size at mostN/t, and three bipartite graphs G1,2, G1,3, G2,3, each Gi,j between Vi and Vj, for which there are at leastε(N/t)³ triangles with one edge in eachG_i,j and less thanρ-proportion of these triangles forms hyperedges inG. By the choice ofG, at least (1−ρ)-proportion of these triangles forms hyperedges inG.

We show that

e^{210ε−2s3/2} ·(1−4ρ)^−4s2 · 16

ε 4s

≤N/t. (4)

Then, by Lemma 8 applied with δ = ε, η = ρ, and m = N/t, the ordered hypergraph G contains a copy ofK₃⁽³⁾(s) as an ordered subhypergraph. By the definition ofG, all hyperedges of such copy are red inχ, which then finishes the proof.

To estimate (4), we estimate each of the three terms in the above expression separately by N^1/3. The exponent in the first term in (4) is

log (e)2¹⁰ε⁻²s^3/2≤2²³·ρ^−30d2·d⁶·s^3/2 ≤ log (N/t)

3 .