Simple juntas for shifted families

We say that a family $\mathcal F$ of $k$-element sets is a $j$-junta if there is a set $J$ of size $j$ such that, for any $F$, its presence in $\mathcal F$ depends on its intersection with $J$ only. Approximating arbitrary families by $j$-juntas with small $j$ is a recent powerful technique in extremal set theory. The weak point of all known approximation by juntas results is that they work in the range $n>Ck$, where $C$ is an extremely fast-growing function of the input parameters, such as the quality of approximation or the number of families we simultaneously approximate. We say that a family $\mathcal F$ is shifted if for any $F=\{x_1,\ldots, x_k\}\in \mathcal F$ and any $G =\{y_1,\ldots, y_k\}$ such that $y_i\le x_i$, we have $G\in \mathcal F$. For many extremal set theory problems, including the Erd\H{o}s Matching Conjecture, or the Complete $t$-Intersection Theorem, it is sufficient to deal with shifted families only. In this note, we present very general approximation by juntas results for shifted families with explicit (and essentially linear) dependency on the input parameters. The results are best possible up to some constant factors, moreover, they give meaningful statements for almost all range of values of $n$. The proofs are shorter than the proofs of the previous approximation by juntas results and are completely self-contained.


Introduction
We say that a family J ⊂ 2 [n] is a j-junta, if there exists a set J ⊂ [n] j and a family J * ⊂ 2 J such that J = {F ⊂ [n] : F ∩ J ∈ J * }. We call J as above the center of the junta and J * the defining family.
We say that F ⊂ 2 [n] is t-intersecting if |F 1 ∩F 2 | ≥ t for any two F 1 , F 2 ∈ F. We say "intersecting" instead of "1-intersecting" for shorthand. Similarly, if for any A ∈ A ⊂ 2 [n] and B ∈ B ⊂ 2 [n] we have |A ∩ B| ≥ t, then we say that A and B are cross t-intersecting. We say "cross-intersecting" instead of "cross 1-intersecting" for shorthand. In a seminal paper [4], the authors proved the following theorem.
Theorem 1 ( [4]). There exist functions j(r), c(r) such that for any integers 1 < j(r) < k < n/2, if A ⊂ [n] k is an intersecting family with |F| > c(r) n−r k−r then there exists an intersecting j-junta J with j ≤ j(r) and This result is, in fact, a corollary of the analogous statement concerning cross-intersecting families.
Theorem 2 ( [4]). There exist functions j(r), c(r) ≥ 1 such that the following holds. Let n, a, b ∈ N with 1 < j(r) < a ≤ b and a + b < n and let A ⊂ such that |A| > c(r) n−r a−r and |B| > c(r) n−r b−r . Then there exist cross-intersecting j-juntas J and I with j ≤ j(r), such that |A \ J | ≤ c(r) · n − r a − r and |B \ J | ≤ c(r) · n − r b − r .
Clearly, these results are meaningful only for n > Ck and n > Cb, respectively, where C = C(r) is sufficiently large (otherwise, say, in the first result, we may have C(r) n−r k−r ≥ n k , in which case the displayed inequality becomes trivial).
Recently, this result was extended by Keller and Lifshitz [14] to the setting of cross-dependent families. We say that F 1 , . . . , F s are cross-dependent if there is no choice F 1 ∈ F 1 . . . , F s ∈ F s such that F i are pairwise disjoint.
Theorem 3 ( [14]). Let s, r be some constants, k < n 2s , and consider cross-dependent families F 1 , . . . , F s ⊂ [n] k . Then there exist C = C(s, r) and cross-dependent C-juntas J 1 , . . . , J s ⊂ [n] k , such that for each i ∈ [s] Again, the theorem only makes sense for n > C ′ (s, r)k, and the dependence of C ′ on s, r is not explicit (and at least exponential in s, r). In many cases, it is desirable to have a control of the behaviour of C ′ (s, r). In particular, this is the case for the Erdős Matching Conjecture (cf. [5], [9], [10]).
We say that the family F is shifted if for any F = {x 1 , . . . , x k } ∈ F and any G = {y 1 , . . . , y k } such that y i ≤ x i , we have G ∈ F. One makes the family shifted by performing shifts (see definition in Section 3.2). We refer to the survey of the first author [8]. Shifting is a very useful combinatorial operation, which preserves many properties of a family. In particular, it preserves the size of each set and the family. Moreover, it preserves the property of being (cross-) t-intersecting, crossdependent etc. Thus, for many extremal problems (including the Erdős Matching Conjecture, the Full Erdős-Ko-Rado theorem etc.), it is sufficient to restrict oneself to shifted families.
The purpose of this note is to show that one can obtain junta-type results for shifted families with essentially best possible dependencies on the parameters with purely combinatorial techniques. 1 In the next section, we illustrate our ideas by giving a simple proof of a stronger version of Theorem 2 for shifted families. In Section 3, we shall give a very general junta approximation-type statement and deduce several of its combinatorial implications, including a stronger version of Theorem 3 for shifted families.

Cross t-intersecting families
Let us first illustrate our methods in the setting of Theorem 2.
a and B ⊂ [n] b . Put j := 2r − t − 1. If n ≥ 2a then there exist cross t-intersecting j-juntas J and I with center [j] such that 1 All previous junta-type theorems rely on results from discrete Fourier analysis.
Proof. For a family F ⊂ 2 [n] and sets X ⊂ S ⊂ [n] we use the following notation: Note that F(X, S) is regarded as a subfamily of 2 [n]\S . The following lemma was proved by the first author (cf., e.g., Proposition 9.3 in [8] Let us put Note that J * and I * may contain the empty set. We claim that J and I are the desired juntas. First, by the definition, it is clear that |A \ J | ≤ 2 j n−j a−r , and similarly for B and I. Second, assume that J * and I * are not cross t-intersecting. This implies that there are two sets X ∈ J * and Y ∈ I * such that |X ∩ Y | ≤ t − 1. Denote x := |X|, y := |Y |. Consider, the families A ′ := A(X, [j]) and B ′ := B(Y, [j]). Note that the former family contains (a − x)-element sets and the latter contains (b − y)-element sets. Due to Lemma 5, A ′ and B ′ are cross t ′ -intersecting, where by the definition of J , I. In particular, x, y ≤ r − 1.
To finish the proof, we need the following lemma, essentially proven in [6].
For the sake of completeness, we shall give the proof of Lemma 6 after the proof of Theorem 4. Applying this lemma to our situation with n ′ := n − j, a ′ := a − x, b ′ := b − y, t ′ := 2r − 1 − x − y, we get that at least one of the following two inequalities is valid: We have x, y ≤ r − 1 and j ≥ r, and thus (a − x) − t ′ = a − r − (r − 1 − y) ≤ a − r. Thus, (4) contradicts (2). But, similarly, (5) contradicts (3). Therefore, we conclude that choosing such X and Y was impossible in the first place. That is, J * and I * , and therefore J and I, are cross t-intersecting.
One may argue that the constants in (1) are still quite bad. Let us derive the following corollary, showing that, at the expense of slightly worse bounds on n and j, one can get rid of the constants. Corollary 7. Fix integers a ≥ b ≥ r > t > 0 and shifted cross t-intersecting families A ⊂ [n] a and B ⊂ [n] b . For any ǫ > 0, put j = 2cr − t − 1, where c := 1 + 2+ǫ 2ǫ 2 log e 4. If n ≥ (2 + ǫ)a then there exists cross t-intersecting j-juntas J and I with center [j] such that Proof. Apply Theorem 4 A, B with cr playing the role of r. Then we get cross t-intersecting juntas I, J with center [j], satisfying (1). To prove the corollary, it is sufficient to show the second inequality in the following chain of inequalities: where k ∈ {a, b}. We have where the second inequality in the first line holds due to the bound on n and the fact that is the biggest if x is as close to 1/2 as possible, and the last inequality holds due to the choice of c.
Proof of Lemma 6. The proof consists of two propositions. Let us say that a set F ⊂ [n] has property t if there exists some i ≥ 0 such that Proof. Interpret each k-element set F as a random walk on where at step i we go one up or one right depending on whether the i'th element belongs or does not belong to F , respectively. Then the sets having property t are exactly the ones that correspond to random walks that hit the line x = y. By reflection principle, the number of such walks is equal to the number of shortest paths from (−t, 0) to (n − k, k − t), which is exactly n k−t .
Proposition 9. Suppose that the shifted families A, B ⊂ 2 [n] are cross t-intersecting. Then either all A ∈ A have property t or all B ∈ B have property t + 1.
Together, these two propositions imply a stronger statement: either

The general statement and its implications
In this section, we state a general junta approximation result and then deduce several corollaries for shifted families satisfying different properties. Let us present a geometric interpretation of the property that we are working with.
First, consider the case of one family F. Imagine that we are given an n + 1 × n + 1 grid (thought of as a subset of the plane) and we are doing a walk on the grid starting from (0, 0) and at each step we can either go from (i, j) to (i + 1, j) or to (i + 1, j + 1). The steps {i 1 , . . . , i s } of the walk at which the random walk goes "diagonally" give us a set in 2 [n] and, obviously, there is a one-to-one correspondence between the subsets in 2 [n] and such walks. The walks with exactly k diagonal steps correspond to k-element subsets of [n]. Thus, any family F corresponds to a collection of walks on this grid.
Next, consider the line λ defined by αy = x + q for some α, q > 0 (here, by x and y we mean the coordinates on the plane). The property that we are interested in is "any walk from F hits λ for some x", where by hitting λ we simply mean that αy ≥ x + q for the corresponding (x, y). 2 As was noted long time ago by the first author [6,7], there is a close relationship between different combinatorial properties of F and the property that F hits λ for appropriately chosen λ. Here, we explore this relationship further. Actually, this relationship is the most transparent for the case of several families (i.e., in the "cross setting").
It is easy to generalize the property to "hit the line" to the case of several families F 1 , . . . , F s . Consider a hyperplane π defined by s i=1 α i y i = x + q for some α i , q > 0. Represent each F i ∈ F i as a walk in the coordinate plane spanned by y i and x. Then we are interested in the property that the "joint walk (represented by the points (x, y 1 , . . . , y s )) hits the hyperplane π".
Let us show how, e.g., the cross t-intersecting property is related to "hitting the plane" property. Consider two families F 1 , F 2 and assume that they hit the plane π" for π defined by y 1 + y 2 = x + t. Then it is easy to see that F 1 and F 2 are cross t-intersecting: indeed, for any F 1 ∈ F 1 , F 2 ∈ F 2 find an integer point p := (x, y 1 , y 2 ) such that the joint walk hits the plane π at p. In terms of sets, it means that |F 1 ∩ [x]| + |F 2 ∩ [x]| ≥ x + t, and by pigeon-hole principle, these two sets intersect in at least t elements, even restricted to [x]. More importantly, if F 1 and F 2 are cross t-intersecting and shifted, then they must hit the plane π (cf. [8] for the easy proof)! Finally, let us describe the idea that is behind the junta approximation result below. For some of the lines, as x grows, it gets increasingly more and more unlikely that a random walk (biased random walk, or random walk with fixed number of diagonal steps) hits this line at point x. Thus, a bulk of the family must cross the line for small x and stay under it for large x. But it is easy to see that this part of the family is an x-junta with the same "hitting the line" property. We have a similar situation for several families, i.e., "sums" of random walks and "hitting a hyperplane" property.
In what follows, all the logarithms have base e.
Theorem 10. Let n, s ≥ 2 and k 1 , . . . , k s be positive integers and q be a non-negative real number. Fix some positive reals α 1 , . . . , α s and a subset of positive integers S.
Fix a positive real r = r(s). Then there exist juntas J 1 ⊂ [n] k 1 , . . . , J s ⊂ [n] ks with center [j] such that for any F i ∈ J i , i ∈ [s] the inequality (7) holds with some ℓ ∈ S ∩ [j] and, moreover, provided one of the three conditions below holds for some real ǫ ∈ (0, 1/3]. (1) We have j : We have j := max i∈[s] α i sr log(e 2 α i s)) and n ≥ max i∈[s] e 2 α i sk i .

Remarks.
• The two first conditions on n, j in the theorem coincide when α 1 k 1 = . . . = α s k s , which is the case in many potential applications. • We note that, unlike the previous results, our junta approximations work for essentially the full range of n. Indeed, one can easily see that in general these problems only make sense for n ≥ i∈[s] α i k i . • We note that, in general, we can get rid of the log factor in the definition of j by putting r = C log −1 (σ/k i ). However, if we want to get an approximation with, say, |F i \J i | ≤ n−1 k i −1 , then cannot remove log(σ/k i ) from Theorem 10 and make both j linear in s and n linear in sk. Indeed, consider the problem of approximating one s-matching-free family and assume that j = Csr and n = C ′ sr for some absolute C, C ′ . (s is sufficiently large.) Clearly, the best center J of size j to take is [j]. Then consider the following s-matching-free family Theorem 11. Let n, s ≥ 2 be positive integers and q be a non-negative real number. Fix some positive real α 1 , . . . , α s and p 1 , . . . , p s ∈ (0, 1). Fix a subset of positive integers S. For i ∈ [s], let F i ⊂ 2 [n] be such that for any F i ∈ F i , i ∈ [s], there exists ℓ ∈ S such that F i satisfy (7).
Fix a positive real number r = r(s). Then there exist juntas J 1 , . . . , J s ⊂ 2 [n] with center [j] such that for any F i ∈ J i , i ∈ [s] the inequality (7) holds with some ℓ ∈ S ∩ [j] and, moreover,  The proof of Theorem 11 is almost the same as that of Theorem 10, moreover, the calculations become easier.
3.1. Combinatorial consequences. The first consequence is for families satisfying a cross-union condition. Note that the case q = 1 corresponds to cross-dependence.
Theorem 12. Let s ≥ 2 be an integer and ǫ ∈ (0, 1/3], r = r(s) be positive real numbers. For i ∈ [s], let F i ⊂ [n] k i be such that |F 1 ∪ . . . ∪ F s | ≤ k 1 + . . . + k s − q for some integer q > 0. Assume that F i are shifted. Then there exist juntas J 1 ⊂ [n] k 1 , . . . , J s ⊂ [n] ks with center [j] such that for each i |F i \ J i | ≤ k i n r n k i and for any F i ∈ J i , i ∈ [s], we have |F 1 ∪ . . . ∪ F s | ≤ k 1 + . . . + k s − q, provided one of the following conditions holds. ( We have j := rs log(e 2 s) and n ≥ max i∈[s] e 2 sk i .
Proof of Theorem 12. The following proposition is a consequence of the shiftedness of F i (see [8]).

Proposition 13. For any
Idea of the proof. Assuming the contrary, it is not difficult to find shifts of F i that would not satisfy Now we can apply Theorem 10 (in one of the three assumptions on j, n) to F i with α 1 = . . . = α s = 1 and S := [n]. This gives us the juntas that satisfy (8). However, it is then clear that for Theorem 14. Let s ≥ 2, t ≥ 1 be integers and ǫ ∈ (0, 1/3], r = r(s) be positive real numbers. For We have j := 3sr and n ≥ 2e 2 k.
Proof. The proof is very similar to the previous one. Due to shiftedness of F i , for any F i ∈ F i , i ∈ [r], we must have Then apply Theorem 10 with q := t s−1 , α = α i := 1 s−1 .
Analogous consequences for non-uniform families can be obtained from the p-biased Theorem 11. Since it is straightforward, we leave the details to the reader.

Why do juntas necessarily have center
Let us put The following is the key property of sets in F ′′ i . Proposition 16. For any F ′′ 1 ∈ F ′′ 1 . . . , F ′′ s ∈ F ′′ s , the property (7) holds for some ℓ ∈ S ∩ [j]. Proof. If not, then it must hold for some ℓ > j. But, by definition of Let us note the following simple facts: . Indeed, by Proposition 16 this inequality holds for any F ′′ 1 ∈ F ′′ 1 , . . . , F ′′ s ∈ F ′′ s . But, by definition, To conclude the proof that J i are the desired juntas, we are only left to show that Proof. Partition F ∈ F ′ i into classes by the maximum value of ℓ, for which |F ∩ [ℓ]| ≥ k i ℓ/σ. Obviously, ℓ must have the form g(t) := ⌊σt/k i ⌋ for integer t ∈ [k i j/σ, k]. Since it does not affect the calculations and simplifies the presentation, in what follows we shall treat σt/k i for each t as an integer (and thus assume that g(t) = σt/k i ). Thus, we have It is not straightforward to bound these terms from above. Thus, we will have to do some auxiliary calculations. In [10, Appendix, (52)], we showed the following useful inequality, valid for any a, b > 0 and integer r 1 ≤ a, r 2 ≤ b: Using (10), we can obtain the following bound for each term in (9).
For positive a, b and x ∈ (0, 1), the function x a (1− x) b is increasing for x < a/(a+ b) and decreasing for x > a/(a + b). We claim that, for n ≥ (1 − ǫ) −1 σ, the function f t (n) · (n/k i ) r is decreasing. Due to the last remark, we only have to show that t−r where the second inequality is due to the fact that j ≥ σr ǫk i . Thus, to show that f t (n) · (n/k i ) r ≤ c for some c > 0 and n ≥ (1 − ǫ) −1 σ, it is sufficient to show the same inequality for n = (1 − ǫ) −1 σ. In that case, we have But this is exactly the probability that the sum X of g(t) i.i.d. indicator random variables with the probability of success equal to (1−ǫ)k i σ is equal to t. Thus, We have E X = (1−ǫ)t. Recall the following Chernoff-type inequality for a sum of m i.i.d. indicator random variables with probability of success p and a > 0 (cf., e.g., [2,Theorem A.1.11]): Recall that ǫ ≤ 1/3. We get that Using the last displayed bound above, we get that where the last bound is due to the fact e −x ≤ 1 − x/2 for x < 1. Due to our choice of j, we get that and therefore k t=k i j/σ f t ·(k i /n) r ≤ 1 for n = (1−ǫ) −1 σ. As we have already mentioned, this implies that the same inequality holds for f t (n) for any n ≥ (1 − ǫ) −1 σ. Combining everything together, This concludes the proof of the proposition.
The second part of Theorem 10 is proven analogously, with the only difference that we define F ′ i , F ′′ i as follows: and show that On the other hand, Combining the two bounds above, we obtain (eα i s) r e r+r log(α i s) = 1. 4.2. Theorem 11. The first part of the proof repeats word for word the proof of Theorem 10, with an obvious replacement of [n] k by 2 [n] in the definition of J i . The only difference is that we need another version of Proposition 16: With the same definition of g(t) (and the same convention concerning omitting integer parts), we get that The remainder of the proof is virtually identical to the part of the proof of Proposition 16 after the definition of f t (n). Recall that p i ≤ p ′ i := (1−ǫ)p i σ by assumption. Due to the properties of the function x a (1 − x) b and the choice of j, it is sufficient for us to prove the statement for p i = p ′ i . From now on, the proof repeats the proof of Proposition 16 word for word.
The proof goes similarly for the second and the third part of the theorem.

Families with no cross-matching
As an application of Theorem 12, we prove an exact bound concerning cross-dependent families. The following question was addressed by Aharoni and Howard [1], as well as by Huang, Loh and Sudakov [11]. Given F 1 . . . , F s ⊂ [n] k that are cross-dependent, 3 find min i∈[s] |F i |. (We note here that some authors use the term "F 1 , . . . , F s contain a rainbow matching" to refer to the situation, opposite to "cross-dependence".) In [11], the authors proved the following result.
Theorem 19. If n > 3sk 2 and F 1 , . . . , F s ⊂ [n] k are cross-dependent then Later, a version of this theorem for families with different uniformity was obtained in [16]. They have also managed to obtain an inequality of the product of sizes of cross-dependent families, partially answering one of the questions of [11]. In [12], the authors obtained a similar result for a stronger notion of a rainbow matching.
As it is easily guessed from the formula, the families attaining the bound are (13) was obtained for n > f (s)k with some unspecified and very fast growing function f (s) by Keller and Lifshitz in [14] as an application of the junta method. We recommend the reader to consult [14] for more advanced applications of the junta method. The proof of the next theorem follows the framework proposed by the authors of [14]. Our junta approximation gives an almost linear bound.
Theorem 20. The statement of Theorem 19 holds for n ≥ 12ks log(e 2 s).
Proof. Since shifting maintains cross-dependence (cf. [8]), we may assume that F i , i ∈ [s], are shifted. Let us apply Theorem 12 part (3) with q = 1, k 1 = . . . = k s = k and r = 3. We get crossdependent j-juntas J 1 , . . . , J s with j = 3s log(e 2 s) and such that |J i \ F i | ≤ k 3 n 3 n k . In particular, if min i∈[s] |F i | ≥ n k − n−s+1 k , then min i∈[s] |J i | ≥ n k − n−s+1 k − k 3 n 3 n k . The proof consists of two steps. The first step is to show that the juntas must have the structure of the (conjectured) extremal family. We will need the following lemma proved by Huang, Loh and Sudakov [11].
Lemma 21. Let G 1 ⊂ X t 1 , . . . , G l ⊂ X t l be cross-dependent and l i=1 t i ≤ |X|. Then there exists i such that |G i | ≤ (l − 1) |X|−1 t i −1 . Let us use the following notation: for any family G ⊂ 2 Proof. Consider the defining family J * i of J i . If J * i contains l singletons, then these singletons must be 1, . . . , l due to shiftedness. Clearly, if each J * i contain s − 1 singletons, then they cannot contain any sets disjoint to [s − 1], and the claim is proved.

Concluding Remarks
We believe that Theorem 20 is only one example of the many possible scenarios where the shifted juntas enable one to get better, concrete bounds which seem to be elusive for the general juntas obtained by means of discrete Fourier transform methods.
We note that the validity of Theorem 20 for n > Csk with some large C was announced by Keevash, Lifshitz, Long, and Minzer as a consequence of general sharp threshold-type results. In particular, they prove the following strengthening of Theorem 3.
Theorem 25. For each ǫ > 0, there exist constants C 1 , C 2 such that if n > C 1 ks, and F 1 , . . . , F s ⊂ [n] k are cross-dependent families, then there exist cross-dependent (C 2 s)-juntas J 1 , . . . , J s on the same center, such that |F i \ J i | < ǫs n−1 k−1 for each i.