Rainbow Ramsey problems for the Boolean lattice

We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$.


Introduction
In this paper we consider rainbow Ramsey-type problems for posets. Given posets P and Q, we say that X ⊆ Q is a weak copy of P if there is a bijection α : P → X such that p ≤ P p ′ implies α(p) ≤ Q α(p ′ ). If α has the stronger property that p ≤ P p ′ holds if and only if α(p) ≤ Q α(p ′ ), then X is a strong or induced copy of P . A copy X of P is monochromatic with respect to a coloring c : Q → N if c(q) = c(q ′ ) for all q, q ′ ∈ X and rainbow if c(q) = c(q ′ ) for all q, q ′ ∈ X. We will be looking for monochromatic and/or rainbow copies of some posets in the Boolean lattice B n , the subsets of an n-element set ordered by inclusion. The set of elements of B n corresponding to sets of the same size is called a level of B n . Definition 1.1. The weak Ramsey number R(P 1 , P 2 , . . . , P k ) is the smallest number n such that for any coloring of the elements of B n with k colors, say 1, 2, . . . , k there is a monochromatic copy of the poset P i in color i for some 1 ≤ i ≤ k. More formally, any coloring c : B n → [k] admits a weak copy of P i in color i for some i. We simply write R k (P ) for R(P 1 , P 2 , . . . , P k ), if P 1 = . . . = P k = P . We define the strong Ramsey number R * (P 1 , P 2 , . . . , P k ) and R * k (P ) for strong copies of posets analogously.
Ramsey theory of posets is an old and well investigated topic, see e.g., [11,14]. However the study of Ramsey problems in the Boolean lattice was initiated only recently: Weak Ramsey numbers were studied by Cox and Stolee [4] and strong Ramsey numbers were investigated by Axenovich and Walzer [2].
In this article we study rainbow Ramsey numbers for the Boolean lattice. We call a coloring of a poset rainbow, if each element of the poset has a different color. Definition 1.2. For two posets P, Q the weak (or not necessarily induced) rainbow Ramsey number RR(P, Q) is the minimum number n such that any coloring of B n admits either a monochromatic weak copy of P or a rainbow weak copy of Q. Strong (or induced) rainbow Ramsey number can be defined analogously and is denoted by RR * (P, Q).
Rainbow Ramsey numbers for graphs have been intensively studied (they are sometimes called constrained Ramsey numbers or Gallai-Ramsey numbers), for a recent survey see [5]. The results on the rainbow Ramsey number for Boolean posets are sporadic [3,10]. Nevertheless, the following easy observation connects the (usual) Ramsey numbers to the rainbow Ramsey numbers. Proof. To see (i) observe that if a coloring c uses at most |Q| − 1 colors, then clearly it cannot contain a rainbow weak copy of Q. Therefore any such coloring showing R |Q|−1 (P ) > n also shows RR(P, Q) > n. An identical proof with strong copies implies (ii).
In Section 2, we show an example of posets P, Q for which (ii) of Proposition 1.3 holds with strict inequality. Unfortunately, we do not know whether (i) holds with strict inequality for some posets P, Q.
Many of the tools used in [2,4] come from the related Turán-type problem, the so-called forbidden subposet problem. Let us introduce some terminology. For a poset P , a family F ⊆ B n of sets is called (induced) P -free if F does not contain a weak (strong) copy of P . The size of the largest (induced) P -free family in B n is denoted by La(n, P ) (La * (n, P )). For a poset P we denote by e(P ) the maximum number m such that for any n the union of any consecutive m levels is P -free. The analogous strong parameter is denoted by e * (P ). The most widely believed conjecture [6] in the area of forbidden subposet problems states that for any poset P we have lim n→∞ La(n, P ) n ⌊n/2⌋ = e(P ) and lim n→∞ La * (n, P ) n ⌊n/2⌋ = e * (P ).
It is worth noting that this conjecture is already wide open for a very simple poset called the diamond poset D 2 (defined on four elements a, b, c, d with relations a < b, c < d). See [9] for the best known bounds in this direction.
For a family F ⊆ B n of sets, its Lubell-mass is λ n (F ) = F ∈F . For a poset P , we define λ n (P ) to be the maximum value of λ n (F ) over all P -free families F ⊆ B n and λ max (P ) is defined to be sup n λ n (P ). Its finiteness follows from the fact that every poset Q is a weak subposet of P |Q| (where P l denotes the l-chain, the totally ordered set of size l) and the k-LYM-inequality stating that λ n (F ) ≤ k for any P k+1 -free family F ⊆ B n . Analogously, λ * n (P ) is the maximum value of λ n (F ) over all induced P -free families F ⊆ B n and λ * max (P ) is defined to be sup n λ * n (P ). It was proved to be finite by Méroueh [12].
We say that a poset is uniformly Lubell-bounded if e(P ) ≥ λ n (P ) holds for all positive integers n. Similarly, a poset is uniformly induced Lubell-bounded if e * (P ) ≥ λ * n (P ) holds for all positive integers n. For k ≥ 2 the generalized diamond poset D k consists of k+2 elements a, b 1 , b 2 , . . . , b k , c with a being the smallest element, c being the largest element and the b i 's forming an antichain. Griggs, Li and Lu [7] proved that infinitely many of the D k 's are uniformly Lubell-bounded and Patkós [13] proved that an overlapping but distinct and infinite subset of the D k 's is uniformly induced Lubell-bounded.
In [2] and [4] it was observed that if P is uniformly Lubell-bounded or uniformly induced Lubell-bounded, then R k (P ) = k · e(P ) and R * k (P ) = k · e * (P ) holds respectively. Our main result concerning weak rainbow Ramsey numbers extends the above observation. Theorem 1.4. Let P be a uniformly Lubell-bounded poset and F ⊆ B n be a family of sets with λ n (F ) > e(P )(k − 1). Then any coloring of c : F → N admits either a monochromatic weak copy of P or a rainbow copy of P k .
Note that the lower bound of Corollary 1.5 follows simply from using |Q| − 1 colors, and so avoiding rainbow weak copies of Q, to color e(P ) consecutive levels by each color, and so avoiding monochromatic weak copies of P .
The analogous coloring for strong copies yields a lower bound RR * (P, Q) = e * (P )(|Q| − 1), but one can easily observe that in most cases this trivial coloring can be improved: If Q does not have a smallest element, then one can color ∅ by an otherwise unused color i. Since no other sets are colored i it does not help creating a monochromatic copy of P , and since Q does not have a smallest element, it does not help creating a strong rainbow copy of Q. Therefore one can introduce the following function. For any poset Q let f (Q) = 0 if Q has a largest and a smallest element, let f (Q) = 2 if Q has neither a largest nor a smallest element, and let f (Q) = 1 otherwise. One obtains RR * (P, Q) ≥ e * (P )(|Q| − 1) + f (Q) for all posets P, Q. Question 1.6. For which uniformly induced Lubell-bounded posets P is it true that RR * (P, Q) = e * (P )(|Q| − 1) + f (Q) holds for any poset Q?
We will show that the chain of length two P 2 does not possess the above property. Otherwise we will mostly study the case of Q being the antichain A k of size k. Theorem 1.7. For any poset P with λ * max (P ) ≥ 2, we have In particular, for every uniformly induced Lubell-bounded poset P we have We obtain the following general upper bound.
Theorem 1.8. For any integer k ≥ 2 let m k = min{m : m ⌊m/2⌋ ≥ k}. Then we have Observe that m k = Θ(log k) and we were not able to improve on this gap between the lower and upper bound even for chains. In particular, the value of RR * (P 2 , A k ) is still unknown. It is between k + 2 and k + m k .
Most of our proofs will go along the following lines: suppose c is a coloring using at least C ≥ k colors that does not admit a rainbow A k , then the union of some C − k + 1 color classes is "small". Therefore it is natural to investigate the following four functions that seem to be interesting in their own right. Definition 1.9. F (n, k) is the smallest integer m such that any k-coloring c : B n → [k] admits a rainbow copy of A k provided every color class is of size at least m. G(n, k) is the infimum of all reals γ such that any k-coloring c : B n → [k] admits a rainbow copy of A k provided every color class has Lubell-mass at least γ.
F ′ (n, k) and G ′ (n, k) are defined by changing coloring to partial coloring (i.e. we only color some subset of the elements of B n ) in the definition of F (n, k) and G(n, k).
Structure of the paper. The remainder of the paper is organized as follows: Section 2 contains some easy observations, preliminary results and their immediate consequences. Theorem 1.4 and other results on weak copies are proved in Section 3. Results on the four functions F , F ′ , G and G ′ are shown in Section 4, and Section 5 contains the proofs of Theorem 1.7 and Theorem 1.8.
We use n ≤k to denote k j=0 n j . All logarithms are of base 2 in this paper.

Preliminaries
In this section we gather some auxiliary results. We start with calculating the Lubell mass of subcubes of B n . Proof.
Here we use the equation n−b i=a i a n−i b = n+1 a+b+1 . This can be proved the following way. The right hand side denotes the number of ways to pick an (a + b + 1)-element subset {x 1 , . . . , x a+b+1 } of [n + 1] with x 1 < x 2 < · · · < x a+b+1 . Let us assume x a+1 = i + 1. Then i is between a and n − b, there are i a ways to pick {x 1 , . . . , x a } and n−i b ways to pick {x a+2 , . . . , x a+b+1 }.
The lower bounds in most of our theorems are obtained via trivial colorings where sets of the same size receive the same color. Let us introduce the following parameters: let m(P ) = max{m : B m does not contain a weak copy of P } and m * (P ) = max{m : B m does not contain a strong copy of P }. We say that Q ⊂ B n is thin if Q contains at most one set from each level. Also, let r * (P ) = max{r : B r does not contain a thin, strong copy of P }. Note that the corresponding weak parameter r(P ) = max{r : B r does not contain a thin, weak copy of P } trivially equals |P | − 2 as B |P |−1 contains a chain of length |P | and thus a weak copy of P .
In the next proposition we prove some lower bounds using non-trivial colorings. A poset P is said to be connected if for any pair p, q ∈ P there exists a sequence r 1 , r 2 , . . . , r k such that r 1 = p, r k = q and r i , r i+1 are comparable for any i = 1, 2, . . . , k − 1. Observe that c and c * do not admit a weak rainbow copy of Q as only |Q| − 1 colors are used. By definition of m(P ), for any set T ⊆ R the family F T = {F ⊆ N : F ∩ R = T } cannot contain a weak copy of P . Thus a monochromatic weak copy of P (admitted by c) must contain two sets F, F ′ with F ∈ F T and F ′ ∈ F T ′ such that |T | = |T ′ | and T = T ′ . As P is connected, we can choose F, F ′ to be comparable. However, since any F ∈ F T is incomparable to any F ′ ∈ F T ′ as T is incomparable to T ′ , a contradiction. So the coloring c does not admit a monochromatic weak copy of P . This proves (i), and one can prove (ii) in a similar way.
To see (iii) let us consider the trivial coloring c : B r * (Q) → {0, 1, . . . , r * (Q)} defined by c(F ) = |F |. As P is connected with |P | ≥ 2, c does not admit a monochromatic copy of P and by definition of r * (Q), c does not admit a rainbow strong copy of Q.
Proof. Let F ⊂ B n be a thin antichain. Then we claim |F | ≤ n − 2 holds, which shows Also, if a 1-element and an (n − 1)-element set are in F , they have to be complements, and then no other sets can be in F .
For the upper bound we prove the stronger statement that B n contains a thin antichain of size n − 2 with f n−1 = |F ∩ [n] n−1 | = 0. We proceed by induction on n. The statement is trivial for n = 4 and n = 5. Let us assume the statement holds for n, and prove it for n + 2. Hence we can find a thin antichain F in B n that has cardinality n − 2 and does not contain a set of size is a thin antichain of size n without an (n + 1)-element set.

Proposition 2.2 and Proposition 2.3 together yield RR
showing that P 2 does not possess the property of Question 1.6 and that there exists a pair of posets for which Proposition 1.3 (ii) holds with a strict inequality.

Weak copies
In this section we prove Theorem 1.4 and some other results on weak Ramsey and rainbow Ramsey numbers. We start with a couple of definitions.
Let us denote by C n the set of all maximal chains in B n . For a family F ⊆ B n and set F ∈ F we define C n,F to be the set of those maximal chains C ∈ C n for which the largest set of F ∩ C is F . Then the max-partition of C n consists of {C n,F : F ∈ F } and if there is a maximal chain C that is disjoint with F , then we gather these maximal chains into C n,− .
The Lubell mass λ n (F ) is the average number of sets of F in a maximal chain C chosen uniformly at random from C n . As observed by Griggs and Li [8] if we condition on the largest set F in F ∩ C, then we obtain Proof of Theorem 1.4. We proceed by induction on k. The base case k = 1 is trivial as any colored set forms a "rainbow" copy of P 1 . Suppose the statement is proven for k − 1 and let F ⊆ B n be a family of sets with λ n (F ) > e(P )(k − 1). Let us fix a coloring c : F → N and let us consider the max-partition {C n,F : F ∈ F }. Using Applying our inductive hypothesis to D F \ F 1 we either obtain a monochromatic weak copy of P or a rainbow copy of P k−1 . As all sets in D F \ F 1 are colored differently than F , we can extend the rainbow copy of P k−1 to a rainbow copy of P k by adding F .
Remark. Note that a simple modification of the above proof shows that if P is a uniformly Lubell-bounded poset and F ⊆ B n is a family of sets with λ n (F ) > e * (P )(k − 1), then any coloring of c : F → N admits either a monochromatic strong copy of P or a rainbow copy of P k , and therefore RR * (P, P k ) = e * (P )(k − 1) holds.
For r ≥ 2 the poset ∨ r consists of a minimal element and r other elements that form an antichain. Similarly, for s ≥ 2 the poset ∧ s consists of a maximal element and s other elements that form an antichain.
Proof. Any coloring c : B n → N with ||c|| ≥ r + 1 admits a rainbow weak copy of ∨ r : the empty set and one representative from at least r other color classes. Similarly, any coloring c : B n → N with ||c|| ≥ s + 1 admits a rainbow weak copy of ∧ s : the set [n] and one representative from at least r other color classes.
Let us now focus on k-colorings of B N avoiding monochromatic weak copies of ∨ r and for simplicity let us write f k (r) = R k (∨ r ). A simplest construction of such coloring is to color sets of the same size with the same color, and color classes should consist of consecutive layers. Formally, let i 1 , i 2 , . . . , i k be positive integers with k j=1 i j = N + 1 and consider the coloring The empty sum equals 0, so c(F ) = 1 if and only if |F | < i 1 holds.) Let us call such a coloring c consecutive layer k-coloring and let us define g k (r) to be the smallest integer N such that any consecutive layer k-coloring of B N admits a monochromatic weak copy of ∨ r . By definition we have , the binary entropy function. Note that for c ∈ [0, 1] and n large enough we have Note that Cox and Steele [4] obtained general but not tight upper bounds on the Ramsey number R(∨ r 1 , . . . , ∨ rs , ∧ r s+1 , . . . , ∧ rt ). We improve on their result and determine the asymptotics in case all target posets are the same. In the proof we omit floor and ceiling signs for simplicity.
Proof. The proof is based on the following simple observations. Claim 3.3. For any k and r we have ≤a > r} and let us consider a coloring c : B N → [k + 1]. Without loss of generality we may assume c(∅) = k + 1. Assume first that there exists a set F ∈ B N with |F | ≤ min{a : a+f k (2r−1) ≤a > r} and c(F ) = k + 1. Then consider the k-coloring c ′ : If its color is not c(F ), then C is a monochromatic weak copy with respect to c. If the color of C is c(F ) and C contains at least r sets that are colored k + 1 with respect to c, then together with the empty set, they form a monochromatic weak copy of ∨ r with respect to c. Otherwise C contains at least r + 1 sets that were colored c(F ) with respect to c. Note that one of these r + 1 sets may be F . Even then, together with F they form a monochromatic weak copy of ∨ r with respect to c.
Assume next that all sets of size at most min{a : a+f k (2r−1) ≤a > r} are colored k + 1. Then the empty set and r other of them form a monochromatic weak copy of ∨ r . This proves (i).
To prove (ii) let us consider a consecutive layer k-coloring c : B g k (r)−1 → [k] defined by the positive integers i 1 , i 2 , . . . , i k such that c does not admit a monochromatic weak copy of ∨ r . We "add max{a : a+g k (r) ≤a ≤ r} + 1 extra levels", i.e. we let j 1 = max{a : a+g k (r) ≤a ≤ r} + 1, and j h+1 = i h for all 1 ≤ h ≤ k and set N ′ := k+1 h=1 j h − 1. We claim that the corresponding consecutive layer (k + 1)-coloring c ′ does not admit a monochromatic weak copy of ∨ r which proves (ii). Indeed, by definition the union of the first j 1 layers does not contain r + 1 sets, so no monochromatic ∨ r exists in this color. To see the ∨ r -free property of the other color classes observe that for any set F of size j 1 the cube B F,[N ′ ] has dimension g k (r) − 1 and the consecutive layer k-coloring that we obtain by restricting c ′ to B F,[N ′ ] is isomorphic to c. If G is the set corresponding to the bottom element of a copy C of ∨ r , then for a j 1 -subset F of G, the copy C belongs to B F,[N ′ ] , so it cannot be monochromatic.
To prove the theorem we proceed by induction on k. If one can use only one color, then all colorings are consecutive layer 1-colorings and B N does not admit a monochromatic ∨ r if and only if 2 N ≤ r, so g 1 (r) = f 1 (r) = ⌊log r⌋ + 1 and c 1 = 1.
Assume now that the statement of the theorem is proved for k and let us fix ε > 0. Observe that using Claim 3.3 (ii) and the inductive hypothesis we obtain that for r large enough we have where (c k − ε) log r ≤ g k (r) ≤ (c k + ε) log r. We claim that if d k is the constant that satisfies (d k +c k )h( d k d k +c k ) = 1, then the maximum in the above expression is at least (d k −ε) log r. Indeed, there exist positive constants C and δ such that where for the second inequality we used d k < c k and for the penultimate inequality we used that the entropy function is strictly increasing in (0, 1/2). Therefore, we have g k+1 ≥ (c k + d k − 2ε) log r.
On the other hand, according to Claim 3.3 (i), we have By the inductive hypothesis, for large enough r we have We claim that the minimum in the above expression is at most (d k + ε) log r. Indeed, for some positive δ ′ and large enough r we have Therefore, we have f k+1 (r) ≤ (c k + d k + 3ε) log r and consequently 4 F (n, k), F ′ (n, k), G(n, k) and G ′ (n, k) Let us start with the following simple observation that connects the four functions.
Proof. The inequalities F (n, k − 1) ≤ F ′ (n, k − 1) and G(n, k − 1) ≤ G ′ (n, k − 1) are immediate as any coloring is a special partial coloring. If F ′ (n, k − 1) ≤ 2 n k , then let c ′ be any partial coloring of B n that does not admit a rainbow A k−1 . Let c be the k-coloring of B n obtained from c ′ by adding color k to all sets not colored by c ′ . If this last color class is at least the size of the smallest color class of c ′ , say color k − 1, then we are done. If not, then we can recolor some of the sets from color classes 1 to k − 2 to color k such that all color classes have size at least the size of color k − 1 to obtain a coloring c * . (Here we use the assumption F ′ (n, k − 1) ≤ 2 n k .) As c ′ does not admit a rainbow A k−1 , c * does not admit a rainbow A k .
We say that the families F 1 , F 2 , . . . , F l are mutually comparable if for any F i ∈ F i and F j ∈ F with 1 ≤ i = j ≤ l we have F i ⊆ F j or F j ⊆ F i . A simple construction of mutually comparable families is the following. We take a chain ∅ = C 0 ⊆ C 1 ⊆ C 2 ⊆ · · · ⊆ C m = [M], and let H j = {H : C j−1 H C j }. Let each family F i be the union of some H j 's and {C j }'s such that no H j belongs to two different families. We say that the chain ∅ = C 0 ⊆ C 1 ⊆ C 2 ⊆ · · · ⊆ C m = [M] is a core chain of the families F i . Note that the core chain is not necessarily unique.
The next simple lemma is more or less due to Ahlswede and Zhang [1], we include the proof for completeness. . By its maximum size, F belongs to F ′ i . Observe that all sets H ∈ F \ F i must be contained in C := ∩ F ′ ∈F ′ i F ′ by the mutual comparable property and obviously every F ′ ∈ F ′ i contains C. So we can apply induction to F 1 , . . . , (i) F ′ (n, 2) = 2 ⌊n/2⌋ + 2 if n ≥ 5 is odd, and F ′ (n, 2) = 2 ⌊n/2⌋ if n is even.
To obtain the upper bound observe that a partial coloring c : B n → {1, 2} does not admit a rainbow A 2 if and only if the color classes c −1 (1) and c −1 (2) are mutually comparable. Therefore applying Lemma 4.2 to c −1 (1) and c −1 (2) we obtain a core chain ∅ = C 0 ⊂ C 1 ⊂ · · · ⊂ C l ⊂ C l+1 = [n] with ∪ l j=0 B C j ,C j+1 containing both color classes such that for every j the truncated subcube B − C j ,C j+1 contains sets only from one color class. Let d j = |C j+1 \ C j | and thus l j=0 d j = n. Then the color class 1 is not larger, than 2 d j 1 + 2 d j 2 + . . . + 2 d j i + k − 1 where the j h 's are the indices of the subcubes containing only sets of color 1 and k is the number of C j 's of color 1 with both B − C j−1 ,C j and B − C j ,C j+1 containing only sets of color 2. As for positive integers x, y we have 2 x + 2 y ≤ 2 x+y − 2 (unless x = y = 1), to maximize the minimum size of the color classes we must have only two subcubes in the partition. A simple case analysis based on the size of C 1 and the color of ∅, C 1 and [n] finishes the proof of (i).
To prove (ii) let ∅ = C 1 , C 1 , . . . , C j = [n] be the core chain and let us write d h = |C h \ C h−1 | for any 1 ≤ h ≤ j. Let us give all the details of this case analysis. If there exists a d j > ⌈n/2⌉, then the the sum of the d i 's corresponding to the other color class is at most ⌊n/2⌋ − 1, so the color class has size at most 2 ⌊n/2⌋−1 + 2. As no C i has size ⌊n/2⌋ or ⌈n/2⌉, there can be at most one d j with ⌊n/2⌋ ≤ d j ≤ ⌈n/2⌉. If there is such a d j , then the d i 's belonging to the other color sum up to at most ⌊n/2⌋ + 2 and all of them are at most ⌊n/2⌋ − 1. This yields that the size of this color class is at most 2 ⌊n/2⌋−1 + 4. (Note that this is sharp if n is odd, d 1 = ⌊n/2⌋ − 1, d 2 = ⌊n/2⌋, d 3 = 2.) Finally, suppose that all d j 's are at most ⌊n/2⌋ − 1. Note that there are at most 2 d j 's larger than n/3. If all d i 's belonging to one color class are at most ⌊n/2⌋ − 2, then one of the color classes has size at most 2 ⌊n/2⌋−1 . If both color classes have a d i that equals ⌊n/2⌋ − 1, then the remaining d i 's sum to 2 or 3 (depending on the parity of n), so one of the color classes have size at most 2 ⌊n/2⌋−1 + 3.
To prove (iii) first we show that lim inf G ′ (n, 2) ≥ √ 2 + 1. Let us fix a set H of size ⌊ n   Proof. If l = 0, then one of the color classes is a subfamily of {∅, [n]} and thus its Lubell mass is at most 2. Now we proceed by contradiction. Suppose |C 1 | ≤ n/2 and |C l | ≥ n/2; then there exists an index 1 ≤ j ≤ l such that |C j | ≤ n/2, |C j+1 | > n/2. Observe that H 1 ∪H 2 ⊆ D C j ∪B C j ,C j+1 ∪U C j+1 holds and one of the color classes is contained in D C j ∪ U C j+1 .
If λ n (B C j ,C j+1 ) > 1 √ n , then by Claim 4.4, we have max{|C j |, n − |C j+1 |} ≥ n 2/3 and thus By Claim 4.5, we can assume without loss of generality that l ≥ 1,  Proof. Let c : B n → [3] be a coloring that does not admit any rainbow A 3 . Let H 1 , H 2 , H 3 denote the three color classes. As every level admits at most two colors, we can assume without loss of generality that H 1 contains at least ( n ⌊n/2⌋ ) 2 sets of size ⌊n/2⌋. Claim 4.7. If |H 2 |, |H 3 | > n(n + 1) + 2 = n 0 + n 1 + n 2 + n n−2 + n n−1 + n n , then H 2 and H 3 are mutually comparable possibly with the exception of some complement set pairs of size 1 and n − 1.
Proof of Claim. First suppose there exists an incomparable pair H 2 ∈ H 2 , H 3 ∈ H 3 with 3 ≤ |H 2 |, |H 3 | ≤ n − 3. Then the number of sets of size ⌊n/2⌋ that are comparable either to H 2 or to H 3 is at most 2 n−3 ⌊n/2⌋ < ( n ⌊n/2⌋ ) 2 . Therefore, there exists a set H 1 ∈ H 1 of size ⌊n/2⌋ such that H 1 , H 2 , H 3 form an antichain. This contradicts that c does not admit a rainbow copy of A 3 .
Let H ′ 2 ⊆ H 2 , H ′ 3 ⊆ H 3 be maximal mutually comparable subfamilies such that they contain all sets from H 2 and H 3 of size between 3 and n−3. (By the above paragraph, there is such a pair.) By the assumption on |H 2 | and |H 3 |, there exist H 2 ∈ H 2 , H 3 ∈ H 3 with 3 ≤ |H 2 |, |H 3 | ≤ n − 3 and H 2 ∈ H ′ 2 , H 3 ∈ H ′ 3 . This shows that Lemma 4.2 applied to H ′ 2 and H ′ 3 must yield a core chain C 1 , C 2 , . . . , C j that contains a set C i with 3 ≤ |C i | ≤ n − 3. As all sets of size ⌊n/2⌋ that are colored 2 or 3 must be comparable to C i , we obtain that the number of sets of size ⌊n/2⌋ colored 1 is at least n ⌊n/2⌋ − n−3 ⌊n/2⌋ . Suppose finally that there exist incomparable sets H ∈ H 2 , H ′ ∈ H 3 such that H, H ′ are not complement pairs of size 1 and n − 1. We claim that there must exist an ⌊n/2⌋-set H ′′ colored 1 that is incomparable to both H and H ′ contradicting the fact that c does not admit a rainbow A 3 . Indeed, the two worst case scenarios are • if |H| = |H ′ | = 1 or |H| = |H ′ | = n − 1, then the number of ⌊n/2⌋-sets that are not comparable to H or H ′ is n ⌊n/2⌋ − n−2 ⌊n/2⌋ < n ⌊n/2⌋ − n−3 ⌊n/2⌋ , • if |H| = 1, |H ′ | = n − 2, then then the number of ⌊n/2⌋-sets that are not comparable to H or H ′ is n ⌊n/2⌋ − n−2 ⌊n/2⌋−1 < n ⌊n/2⌋ − n−3 ⌊n/2⌋ .
With Claim 4.7 in hand, we are ready to prove (i). To prove F (n, 2) ≤ F ′ (n, 2) we need to show that one of the color classes has size smaller than F ′ (n, 2). If n ≥ 18, then n(n + 1) + 2 < F ′ (n, 2), so if a color class of c has size at most n(n + 1) + 2, then we are done. If two color classes of c are mutually comparable, then by definition, one of these has size less than F ′ (n, 2) and we are done again. By Claim 4.7, the only other possibility is that H 2 , H 3 contains a pair of complement sets {x} ∈ H 2 , [n] \ {x} ∈ H 3 , and there is a maximal mutually comparable pair of subfamilies H ′ 2 , H ′ 3 such that H ′ 2 and H ′ 3 contain all sets from H 2 and H 3 of size between 2 and n − 2. Consider the core chain C 0 , C 1 , . . . , C j of H ′ 2 and H ′ 3 . Let us assume first that no member of the chain C i has size ⌊n/2⌋ or ⌈n/2⌉. Then by (ii) of Corollary 4.3, either H ′ 2 or H ′ 3 has size at most 2 ⌊n/2⌋−1 + 4 and thus one of H 2 , H 3 has size at most 2 ⌊n/2⌋−1 + 2n + 4 < 2 ⌊n/2⌋ ≤ F ′ (n, 2), where the first inequality holds by the assumption n ≥ 18.
Next assume that for any pair We can also assume that n = 2m + 1 is odd as if n is even then it is symmetric to the case x / ∈ C i . Then as all sets of H 2 of size between 2 and n are comparable to [n] \ {x} and all sets of H 2 are comparable to C i , we obtain that H 2 ⊆ D C i \{x} ∪ {[n]}, so |H 2 | ≤ 2 m−1 + 1 < F ′ (n, 2) and we are done.
To show (ii), we need to prove that for any coloring c : B n → [3] that does not admit a rainbow induced copy of A 3 , one of the color classes has Lubell mass at most 1 + √ 2 + o(1). Let us first observe that with a little modification the proof of Claim 4.7 works without the assumption |H 2 |, |H 3 | ≥ n(n + 1) + 2 if n is large enough. The assumption was only used to obtain sets H ∈ H 2 , H ′ ∈ H 3 with 3 ≤ |H|, |H ′ | ≤ n − 3, which in turn was used to obtain the bound |H 1 ∩  not containing P so P, H, H ′ would form a rainbow copy of 2 ) P | ≥ ε ′ n, the number of ⌊n/2⌋-sets containing a fixed set of size ε ′ n is o( n ⌊n/2⌋ , so the proof again can be completed as before. The cases i = n − 2, n − 1 are analogous to the cases i = 2, 1.
We have proved that either one of the color classes of c has Lubell mass at most 1 + , thus it is incomparable only to one member of H 2 , namely {x}, which is in D C 1 , a contradiction. For the same reason any set H ∈ H 2 of size between 2 and |C 1 | must belong to D C 1 \R . We obtained that Now we prove a technical lemma, that will help to finish the proof of Theorem 4.6 (ii). Claim 4.8. Suppose we have two real numbers 0 ≤ β ≤ α ≤ 1. Proof.
Having Claim 4.8 in hand, we distinguish two cases according to the value of α (i.e. the size of C 1 ).
With the reasoning of Case I, this time ∅ and [n] must belong to H 3 and therefore we have In this case we are also done by Claim 4.8 (b).
Proposition 4.10. For any integer k ≥ 2 let l k = ⌊ log(k−1) 2 ⌋. Then we have Proof. Fix an integer k. If n is large enough, then we can pick k − 1 sets F 1 , F 2 , . . . , F k−1 of size ⌊n/2⌋ + l k such that |F i ∩ F j | ≤ 0.26n (take sets uniformly at random from the middle level of B n ). Let us define a coloring c by c Observe that c does not admit a rainbow copy of A k as if c(A) = k with A ∈ U i belongs to a rainbow antichain, then color i is missing. Also, for all i = 1, 2, . . . , k − 1 and

Strong copies
Proof of Theorem 1.7. Let N = 2 + 2⌈λ * max (P )⌉ and consider a coloring c : B − N → N and assume indirectly that it does not admit a monochromatic induced copy of P , nor a rainbow copy of A 3 . If ||c|| ≤ 2, then one of the color classes has Lubell-mass strictly larger than λ * max (P ), so by the definition of λ * max , c admits a monochromatic induced copy of P , a contradiction. Therefore, we can assume that ||c|| ≥ 3. Observe that for any set F ∈ H i the families H 1 \ I F , . . . , H i−1 \ I F , H i+1 \ I F , . . . are mutually comparable as otherwise c admits a rainbow induced copy of A 3 .
Claim 5.1. There exists a set F of size N/2 such that at least 2 colors are used on B − N \ (I F ∪ F c(F ) ).
Note that this claim shows us three sets of different colors with at most one containment relation between them. No containment relation would show a rainbow A 3 , thus a contradiction.
Proof of Claim. If F and [N] \ F have different colors, then any set in B − n can be comparable to at most one of them. We know that there is a set G of a third color, without loss of generality F and G are incomparable, thus G and N \ F are both in B − N \ (I F ∪ F c(F ) ) and are of different colors, finishing the proof in this case. So we may assume that for every set F of size N/2 we have c(F ) = c([N] \ F ).
If [N ] N/2 is monochromatic in color i, then for any two sets H, H' from two other color classes, we can pick an F from [N ] N/2 that is not comparable to H and H ′ unless H is a singleton and H ′ is its complement. So either there is a pair of two sets from two other color classes not of this form, or the color class i contains all levels from 2 to N − 2. As N − 3 ≥ e * (P ), we obtain a monochromatic induced copy of P .
N/2 is not monochromatic, then we can pick two sets F, F ′ of size N/2 and of different colors with |F ∩ F ′ | = N/2 − 1. There exists a set G of a third color, we are done unless G is comparable to both F and F ′ . In that case G cannot be comparable to [N] \ F and [N] \ F ′ , thus G, [N] \ F and [N] \ F ′ form a rainbow copy of A 3 , a contradiction.
Let F be a set assured by Claim 5.1. We may assume without loss of generality that c(F ) = 1. Then if c does not admit a rainbow copy of A 3 , then H 2 \I F , H 3 \I F , . . . are mutually comparable families, and at least two of them are non-empty. Because of this, we can apply Lemma 4.2 to obtain a core chain of them ∅ = C 0 ⊆ C 1 ⊆ C 2 ⊆ · · · ⊆ C j = [N].
There is a set C = C i in this chain that has size between 1 and N − 1 with both {G : G ⊆ C} ∩ (∪ ||c|| j=2 H i \ I F ) and {G : G ⊇ C} ∩ (∪ ||c|| j=2 H i \ I F ) being non-empty. This implies F and C are incomparable.
Observe that H i \ I F ⊆ I C for all i = 2, 3, . . . , ||c||. Therefore all sets in B − N \ (I F ∪ I C ) are of color 1. We claim that B − N \ (I F ∪ I C ) contains a copy of B ⌈λ * max (P )⌉ and thus c admits a monochromatic induced copy of P .
Note that we may suppose |C| ≤ N/2. If there exists x ∈ [N] \ (F ∪ C), then we can fix elements y ∈ F, z ∈ C and a set G of size N/2 + 1 with x ∈ G, y, z / ∈ G. Then the cube {H : x ∈ H ⊆ G} is a subfamily of B − N \ (I F ∪ I C ). If F ∪ C = [N], then F = [N] \ C and we can consider elements x, y ∈ F and u, v ∈ C and a set G of size N/2 + 2 with x, u / ∈ G, y, v ∈ G. Then the cube {H : x, u ∈ H ⊆ G} is a subfamily of B − N \ (I F ∪ I C ). Indeed, if not, then one of the color classes on Q S has Lubell-mass strictly larger than λ * max (P ). By the definition of m k , we can pick k subsets S 1 , S 2 , . . . , S k of [m k ] of size ⌊m k /2⌋. As the S i 's form an antichain, the families Q S 1 , Q S 2 , . . . , Q S k are mutually incomparable. By the above paragraph, on each of these families c admits at least k colors otherwise we find a monochromatic induced copy of P . But then we can pick a rainbow antichain from the Q i 's greedily: a set F 1 from Q S 1 , then F 2 from Q S 2 with c(F 1 ) = c(F 2 ) and so on.