Weak Saturation of Multipartite Hypergraphs

Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weaklyH-saturated in F if the edges in E(F)\E(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E(F)\setminus E(G)$$\end{document} admit an ordering e1,…,ek\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_1,\ldots , e_k$$\end{document} so that for all i∈[k]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i\in [k]$$\end{document} the hypergraph G∪{e1,…,ei}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\cup \{e_1,\ldots ,e_i\}$$\end{document} contains an isomorphic copy of H which in turn contains the edge ei\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e_i$$\end{document}. The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.


Introduction
Let F and H be q-uniform hypergraphs (q-graphs for short); we identify hypergraphs with their edge sets.We say that a subgraph G ⊆ F is weakly H-saturated in F if the edges of F \ G can be ordered as e 1 , . . ., e k such that for all i ∈ [k] the hypergraph G ∪ {e 1 , . . ., e i } contains an isomorphic copy of H which in turn contains the edge e i .We call such e 1 , . . ., e k an H-saturating sequence of G in F .The weak saturation number of H in F , wsat(F, H) is the minimum number of edges in a weakly H-saturated subgraph of F .When F is complete of order n, we simply write wsat(n, H).
Weak saturation was introduced by Bollobás [Bol68] in 1968 and is related to (strong) graph saturation: G is H-saturated in F if adding any edge of F \ G would create a new copy of H.However, a number of properties of weak saturation make it a more natural object of study.Firstly, it follows from the definition that any graph G achieving wsat(F, H) has to be H-free (we could otherwise remove an edge from a copy of H in G resulting in a smaller example), while for strong saturation H-freeness may or may not be imposed, resulting in two competing notions (see [MS15] for a discussion).Secondly, a short subadditivity argument originally due to Alon [Alo85] shows that for every 2-uniform H, lim n→∞ wsat(n, H)/n exists.Whether the same holds for strong saturation is a longstanding conjecture of Tuza [Tuz86].And thirdly, weak saturation lends itself to be studied via algebraic methods, thus offering insight into algebraic and matroid structures underlying graphs and hypergraphs.
The most natural case when F and H are cliques was the first to be studied.Let K q r denote the complete q-graph of order r.Confirming a conjecture of Bollobás, Frankl [Fra82], and Kalai [Kal84b,Kal85] independently proved that wsat(n, K q r ) = n q − n−r+q q .Another proof has been given by Alon [Alo85] and in hindsight this conjecture could be also derived from an earlier paper of Lovász [Lov77].While the upper bound is a construction that is easy to guess (a common feature in weak saturation problems), all of the above lower bound proofs rely on algebraic or geometric methods, and no purely combinatorial proof is known to this date.
In the subsequent years weak saturation has been studied extensively [Alo85, Tuz92, EFT91, Pik01a, Tuz88, MS15, Pik01b, Sem97, BS02, Sid07, FG14, BBMR12, BP98,MN18].Despite this, our understanding of weak saturation numbers is still rather limited.For instance we do not know whether for q ≥ 3 we have a similar limiting behavior as in the graph case, in that lim n→∞ wsat(n, H)/n q−1 always exists; this has been conjectured by Tuza [Tuz92].
In this paper we address the case when H = K q r1,...,r d is a complete d-partite q-graph for arbitrary d ≥ q > 1.That is, V (H) is a disjoint union of sets R 1 , . . ., R d with |R i | = r i and in particular, for q = 2 we recover the usual complete multipartite graphs.This is perhaps the next most natural class of hypergraphs to consider after the cliques.
For the host graph F , besides the clique it is natural to consider a larger complete d-partite qgraph K q n1,...,n d .In the latter case we have a choice between the undirected and directed versions of the problem.The former follows the definition of weak saturation given at the beginning, while in the latter we additionally impose that the new copies of H in F created in every step "point the same way", i.e. have r i vertices in the i-th partition class for all i ∈ [d] (see below for a formal definition).
All three above versions have been studied in the past.For q = 2, Kalai [Kal85] determined wsat(n, K r,r ) for large enough n.Kronenberg, Martins and Morrison [KMM21] recently extended it to wsat(n, K r,r−1 ) and asymptotically to all wsat(n, K s,t ).No other values wsat(n, K q r1,...,r d ) are known except for r 1 = • • • = r d = 1 when H is a clique and a handful of closely related cases, e.g., when all r i but one are 1 [Pik01b].When both H and F are complete d-partite, for d = q Alon [Alo85] solved the problem in the directed setting.Moshkovitz and Shapira [MS15], building on Alon's work, settled the undirected case, determining wsat(K d n1,...,n d , K d r1,...,r d ).There has been no progress for d > q.In our main contribution in this paper we settle completely the directed case for all q and d.To state the problem formally, let r = (r 1 , . . ., r d ) and n = (n 1 , . . ., n d ) be integer vectors such that 1 and ⊔ denotes a disjoint union.Let K q n be the complete dpartite q-graph on N whose partition classes are the N i , and let K q r be an unspecified complete d-partite q-graph on the same partition classes, with r i vertices in each N i .Given a subgraph G of K q n , a sequence of edges e 1 , . . ., e k in K q n is a The q-graph G is said to be (directed) weakly K q r -saturated in K q n if it admits a K q r -saturating sequence in the latter.The (directed) weak saturation number of K q r in K q n , in notation w(K q n , K q r ), is the minimal number of edges in a weakly K q r -saturated subgraph of K q n .Theorem 1.1.For all d ≥ q ≥ 2, n and r we have In the above formula [d]  ≤q stands for the set of all subsets of [d] of size at most q, and we use the convention that i∈∅ (n i − r i ) = 1.
As mentioned, the d = q case of Theorem 1.1 was proved by Alon [Alo85].Hence our result generalizes Alon's theorem to arbitrary d ≥ q.When H is balanced, that is when r 1 = • • • = r d , there is no difference between the directed and undirected partite settings.Writing K q (r; d) for K q r,...,r (d times), Theorem 1.1 thus determines the weak saturation number of K q (r; d) in complete d-partite q-graphs.
Corollary 1.2.For all d ≥ q ≥ 2 and n 1 , . . ., n d ≥ r ≥ 1 we have Our proof of Theorem 1.1 combines exterior algebra techniques in the spirit of [Kal85] with a new ingredient: the use of the colorful exterior algebra inspired by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem [BGT21].
Kronenberg, Martins and Morrison ([KMM21], Section 5) remarked that while the values wsat(n, K t,t ) and wsat(K ℓ,m , K t,t ) for ℓ + m = n, which were determined in separate works, are of the same order of magnitude, it is not obvious if there is any direct connection.In our second contribution in this paper we establish such a connection using a tensoring trick.As we have mentioned earlier, 2-graphs H satisfy wsat(n, H) = c H n + o(n), and Alon's proof of this fact [Alo85] can be straightforwardly adjusted to show that wsat(K n,n , H) = c ′ H • 2n + o(n) when H is bipartite.We show that in fact c H = c ′ H .A minor adjustment to our proof gives that, for any rational 0 < α < 1, the quantities wsat(n, H) and wsat(K αn,(1−α)n , H), when αn ∈ Z, are of the same order of magnitude.Setting H = K t,t answers the above question of [KMM21].
For q ≥ 3 while we do not have (yet) the same knowledge of limiting constants, a similar method determines asymptotically the weak saturation number of complete d-partite d-graphs in the clique, generalizing Theorem 4 of [KMM21].
Theorem 1.3.For every bipartite 2-uniform graph H we have Furthermore, for any d ≥ 2 and The rest of the paper is organized as follows.In Section 2 we give a construction for the upper bound in Theorem 1.1.In Section 3 we review the algebraic tools, setting the stage for the lower bound proof in Section 4. In Section 5 we discuss weak saturation in the clique and prove Theorem 1.3.
Notation.As usual, [n] abbreviates the set {1, . . ., n}.The symbol ⊔ denotes a disjoint union of sets.For a set M and integer q ≥ 0, M q and M ≤q denote the set of all subsets of M of size exactly q and of size most q, respectively.We use ± to denote an unspecified factor of either +1 or −1.
K q n denotes the complete q-uniform hypergraph (q-graph) of order n.When the vertex set of the said q-graph is [n], we write K q [n] .The complete d-partite q-graph with n i vertices in the i-th partition class is denoted by K q n1,...,n d ; when n 1 = • • • = n d = n we write simply K q (n; d).Note that in Sections 2-4 we work solely in the directed partite setup (Theorem 1.1), while in Section 5 we deal with the undirected partite and the clique setups (Theorem 1.3).In the directed setup our q-graphs are defined on a vertex set N of size n with a fixed d-partition . Consequently, we use K q n to denote the complete d-partite q-graph on N with respect to this partition.(Up to a graph isomorphism, K q n is uniquely determined by q and n, thus we do not display N in the notation.)For any M ⊆ N the induced subgraph of K q n on M is denoted by . The directed weak saturation number defined above is denoted by w(K q n , K q r ), as opposed to wsat(K q n1,...,n d , K q r1,...,r d ) in the undirected setting, a similar notation was employed in [KMM21].
2 Theorem 1.1: the upper bound In this section we prove the upper bound in Theorem 1.1 by exhibiting a weakly K q r -saturated q-graph G. Fix a subset R ⊆ N such that |R ∩ N i | = r i for every i ∈ [d] and set We define G via its complement in K q n as follows.For every S ∈ Σ choose an edge λ(S) ∈ K q n [R ∪ S] satisfying S ⊆ λ(S).Note that the assignment λ is injective, as λ(S) ∩ (N \ R) = S. Recall that we associate hypergraphs with their edge sets.Define Notice that the choices of λ(S) are not unique, but as the next lemma shows, each of them yields a weakly K q r -saturated q-graph.Such non-uniqueness is a common occurrence in weak saturation: for instance, every n-vertex tree is an extremal example for weak triangle saturation in K n .
Lemma 2.1.The q-graph G defined above is weakly K q r -saturated.Therefore, |T \ R| ≤ k}, and put G −1 := G.We claim that adding any new edge L ∈ K q n with |L \ R| = k to G k−1 creates a new copy of K q r containing L. This gives rise to a K q r -saturating sequence between G k−1 and G k and, by extension, between G = G −1 and G q = K q n .First, notice that G 0 is obtained from G −1 by adding the sole missing edge λ(∅).Doing so creates a new copy of K q r , namely containing L and a fortiori also a new copy of K q r containing L, as desired.

Algebraic background
In this section we introduce the linear algebra tools needed for the proof of the lower bound in Theorem 1.1.In Sections 3.1 and 3.2 we largely follow [Kal84a, Sec.2] though we sometimes provide more detail.(For comparison [Kal85] works with a dual generic basis.We believe that the difference is not essential.)In Section 3.3 we loosely follow [BGT21].
Before we start explaining the algebraic background, we will try to sketch why algebraic tools can be useful in this context.This sketch should be understood loosely-we do not provide any guarantees for the claims in this sketch.In particular, many important technical details are skipped in the sketch.Understanding this sketch is not required in the following text, thus it can be skipped.
Consider first the somewhat trivial case of providing the lower bound on wsat(n, K 3 ), the weak saturation number of the complete graph K 3 in K n .Consider a subgraph G of K n and a saturating sequence e 1 , . . .e k of edges in Because the sequence is saturating, we know that G i contains a copy of K 3 containing e i .This means that the dimension of the cycle space of G i is strictly larger than the dimension of the cycle space of G i−1 .Because the final dimension of the cycle space of K n equals n−1 2 , we may perform at most n−1 2 such steps.In other as required.In the language of algebraic topology (which we however do not use in the proofs, no topological background is required), the property that the dimension of the cycle space increases can be phrased so that a new copy of K 3 in each step belongs to the kernel of the standard boundary operator.For more complicated (hyper)graphs than K 3 it is actually useful to use several independent boundary operators in order to generalize the aforementioned approach.Using such independent operators can be actually efficiently phrased in terms of exterior algebra (without mentioning algebraic topology).They correspond to the left interior product, which we will discuss later on, subject to some suitable independence (genericity) condition. 1

Exterior algebra.
Let N be a set of size n, ordered with a total order <.Later on the elements of N will represent vertices of a q-graph and we will typically denote them by letters such as v or w.Let V be an n-dimensional real vector space with a basis (e v ) v∈N .The exterior algebra of V , denoted by V , is a 2 n -dimensional vector space with basis (e S ) S⊆N and an associative bilinear product operation, denoted by ∧, that satisfies (i) e ∅ is the neutral element, i.e. e ∅ ∧ e S = e S = e S ∧ e ∅ ; (ii For 0 ≤ k ≤ n we denote by k V the subspace of V with basis (e S ) S∈( N k ) .Denote by •, • the standard inner product (dot product) on V as well as on V with respect to the basis (e v ) v∈N and (e S ) S⊆N respectively; that is, for every pair of sets S, T ⊆ N , the inner product e S , e T is 1 if S = T and 0 otherwise.
) is a basis of k V for k ∈ {0, . . ., n}.The formulas (i), (ii) and (iii) remain valid for the basis (f v ) v∈N due to definition of f S and bilinearity of ∧.In particular, V and k V do not depend on the initial choice of the basis.Using (ii) and (iii) iteratively, for S, T ⊆ N we get where sgn(S, T ) is the sign of the permutation of S ∪ T obtained by first placing the elements of S (in our total order <) and then the elements of T .Equivalently, sgn(S, T ) = (−1) α(S,T ) where α(S, T ) = |{(s, t) ∈ S × T : t < s}| is the number of transpositions.
As a consequence we obtain the following useful formula.Let M 1 , . . ., M ℓ be pairwise disjoint subsets of N and s 1 , . . ., s ℓ be integers with 0 ≤ s i ≤ |M i |.Suppose that for each i ∈ [ℓ] we are given Perhaps the closest relation between the boundary operators and the left interior product can be seen in Lemma 3.3 interpreting e R as a simplex with set of vertices R, and f T as an operator removing t times the top-dimensional simplices, yielding a linear combination of simplices f S with r − t vertices.(However, for this relation, it would be even better to express the right hand side using e S so that all possible e S would appear.)Adding a colorful aspect (in our case) then makes it easier to work with multipartite (hyper)graphs rather than complete ones.
for λ Si ∈ R (so that h i ∈ si V ).Then by bilinearity of ∧ and (3) we get Let A = (a vw ) v,w∈N be the transition matrix from (e v ) v∈N to (f v ) v∈N , meaning that f v = w∈N a vw e w .Then, for S ⊆ N of size k, f S can be expressed as where A S|T is the submatrix of A formed by rows in S and columns in T , i.e.A S|T = (a vw ) v∈S,w∈T .As noted in [Kal84a], it follows from the Cauchy-Binet formula that if the basis (f v ) v∈N is orthonormal then (f S ) S⊆N is orthonormal as well.For completeness, we provide a short explanation.Let S, L ⊆ N be a pair of subsets.If |S| = |L|, then f S and f L belong to two orthogonal subspaces of V , namely |S| V and |L| V , and so f S , f L = 0. On the other hand, if |S| = |L| =: k, then by writing f S and f L in the standard basis (e T ) T ⊆N we have that where B t stands for the transpose matrix of B (and expressions like A t L|T stand for (A L|T ) t ), and the last equality holds by the Cauchy-Binet formula (see e.g.Section 1.2.4 of [Gan98]).Notice that for any u ∈ S and w ∈ L we have (A S|N A t L|N ) u,w = f u , f w , and since (f v ) v∈N is orthonormal this is 1 if u = w and 0 otherwise.Therefore, if S = L, the product A S|N A t L|N is the identity matrix and consequently the determinant will be 1.On the other hand, if S = L, the product A S|N A t L|N will have a zero column, and so the determinant will be 0. The above claim follows.
We say that the change of basis from (e v ) v∈N to (f v ) v∈N is generic if det(A S|T ) = 0 for every S, T ⊆ N of the same size; that is, every square submatrix of A has full rank.It is known (see e.g.[Kal84a]) that (f v ) v∈N can be chosen to be both generic and orthonormal.For a basis (f v ) v∈N generic with respect to (e v ) v∈N and a pair of sets S, T ∈ N k we have = 3.2 Left interior product.
The following lemma defines g f , the left interior product of g and f .We refer to Section 2.2.6 of [Ros19] for a more extensive coverage of the topic.
Lemma 3.1.For any f, g ∈ V there exists a unique element g f ∈ V that satisfies To verify that this satisfies (7) let h ∈ V be arbitrary.By bilinearity of •, • and ∧, and orthonormality of (e S ) S⊆N we have To show uniqueness, suppose that z is an element in V that satisfies (7).Then for each T ⊆ N we have e T , z = e T ∧ g, f = e T , g f .
Therefore z and g f are identical, as their inner products with all basis elements coincide.Now assume that f ∈ s V and g ∈ t V , and let S ⊆ N be arbitrary.By (7) we have e S , g f = e S ∧ g, f .
Observe that e S ∧ g ∈ |S|+t while f ∈ s V and these spaces are orthogonal unless |S| + t = s.Hence, It is straightforward to check from the definition that the left interior product is bilinear: and satisfies With sgn(•, •) as defined in Section 3.1 we obtain the following statement.
Lemma 3.2.Let (f v ) v∈N be an orthonormal basis of V .Then, for any S, T ⊆ N we have Proof.Put s := |S| and t := |T |.If t > s then by Lemma 3.1 we have f T f S = 0 and the conclusion follows.So we may assume that s ≥ t, and by the same lemma it follows that f T f S ∈ s−t V .Since the basis (f v ) v∈N is orthonormal, so is the basis (f L ) L∈( N s−t ) of s−t V , as observed in Section 3.1.
Expressing f T f S in this basis and using (7), we obtain Due to (3) and orthonormality of (f v ) v∈N we have f L ∧ f T , f S = 0 unless T ⊆ S and L = S \ T .Therefore, using (3) again we get Lemma 3.3.Let (f v ) v∈N be a generic orthonormal basis of V with respect to (e v ) v∈N .For a pair of sets T, R ⊆ N of sizes t and r, respectively, such that r ≥ t we have where all the coefficients λ S are non-zero.
Proof.By Lemma 3.1 we have that f T e R ∈ r−t V .Since (f S ) S∈( N r−t ) is an orthonormal basis of r−t V , we can write

Applying (7) and (3) gives
Setting λ S = f S ∧ f T , e R for S ∈ N \T r−t , we thus obtain as claimed.In addition, since we assumed that (f v ) v∈N is generic with respect to (e v ) v∈N , we have λ S = ± f S∪T , e R = 0 by (6) for all S ∈ N \T r−t .

Colorful exterior algebra.
As we are interested in multipartite hypergraphs it is natural to assume in addition that the set N is partitioned as a disjoint union Here each N i is ordered by a total order < i .We extend these orders to the whole N as follows, for x ∈ N i and y ∈ N j , we say that x < y if i < j or if i = j and x < i y.
Given the standard basis (e v ) v∈N of V we say that a basis (f v ) v∈N is colorful with respect to this partition if (f v ) v∈Ni generates the same subspace of V = R N as (e v ) v∈Ni for every i ∈ [d]; we denote this subspace V i .Put differently, the transition matrix A from (e v ) v∈N to (f v ) v∈N is a block-diagonal matrix with blocks We also say that (f v ) v∈N is colorful generic (with respect to this partition) if the basis change from (e v ) v∈Ni to (f v ) v∈Ni is generic for every i ∈ [d].It is possible to choose a basis which is simultaneously colorful generic with respect to a given partition and orthonormal by choosing each change of basis from (e v ) v∈Ni to (f v ) v∈Ni generic and orthonormal.By V i we denote the subalgebra of V generated by e S for S ⊆ N i and by k V i the subspace of V i with basis (e S ) S∈( N i k ) ; that is, We claim that the left interior product behaves nicely with respect to a colorful partition.To see this, we first need an auxiliary lemma about signs.
Lemma 3.4.Let U and T be disjoint subsets of N and for all i where c depends only on u 1 , . . ., u d and t 1 , . . ., t d .
Proof.The value sgn(U, T ) is −1 to the number of transpositions in the permutation π of U ∪ T where we first place the elements of U (in our given order on N ) and then the elements of T (in the same order).Considering that for i < j, U i precedes U j and T i precedes T j , the order of the blocks U 1 , . . ., U d , T 1 , . . ., T d in π is (U 1 , . . ., U d , T 1 , . . ., T d ).
By the above, the sign of π ′ equals (−1) c sgn(U, T ).On the other hand, as T i precedes U j for i < j in our order on N , the sign of π ′ is also equal the product sgn(U 1 , T 1 ) • • • sgn(U d , T d ).Equating these two expressions gives the desired identity.
In the following proposition, the f i are not necessarily coming from a colorful generic basis.However, we intend to apply it in this setting.With a slight abuse of notation, we use both for the exterior algebra as well as for the wedge product of multiple elements.(This can be easily distinguished from the context.)Proposition 3.5.Suppose that s 1 , . . ., s d and t 1 , . . ., t d are nonnegative integers with t i ≤ s i ≤ n i for every Proof.We will show that where c comes from Lemma 3.4; in particular, it depends only on t 1 , . . ., t d and s 1 , . . ., s d .By bilinearity of and ∧ it is sufficient to prove (9) in the case when the f i and the h i are basis elements of ti V i and si V i respectively.So, assume for each i ∈ [d] that f i = e Ti and h i = e Si where as required.

Theorem 1.1: the lower bound
In this section we prove the lower bound in Theorem 1.1.Our proof follows a strategy similar to [BBMR12] and [Kal85].Viewing the edges of K q n as elements of the exterior algebra of R N , we will define a linear mapping closely related to the weak saturation process and lower-bound w(K q n , K q r ) by the rank of the corresponding matrix.
As outlined in Section 3, let V be an n-dimensional real vector space with a basis (e v ) v∈N , equipped with a standard inner product •, • with respect to this basis, that is, (e v ) v∈N is orthonormal.Using the exterior product notation of Section 3, define span K q n := span{e T : For an element m ∈ k V the support of m is the set The following lemma, which converts the problem at hand into a constructive question in linear algebra, is analogous to Lemma 3 in [BBMR12].
Lemma 4.1.Let Y be a real vector space and Γ : span K q n → Y a linear map such that for every subset R ⊆ N with |R ∩ N i | = r i for all i ∈ [d] there exists an element m ∈ ker Γ with supp(m) = E(K q n [R]).Then w(K q n , K q r ) ≥ rank Γ. Proof.Suppose the q-graph G 0 is weakly K q r -saturated in K q n and |E(G 0 )| = w(K q n , K q r ).Denote by {L 1 , . . ., L k } a corresponding saturating sequence and by H i a new copy of K q r that appears in and note that Y k = Γ(span K q n ).By assumption, for each i = 1, . . ., k there exist non-zero coefficients {c T : We conclude that Y i = Y i−1 .By repeating this procedure we obtain Our goal now is to define a linear map Γ as in Lemma 4.1.For this purpose let us fix an orthonormal colorful generic basis (f v ) v∈N of V with respect to the partition of N , as described in Section 3.3.Next, We can now state the following auxiliary lemma.
Lemma 4.2.Let z be an integer with d ≥ z ≥ s and let Z ∈ N z .Then Proof.By (10), bilinearity of , and Lemma 3.2 we get The last expression is 0 if |Z ∩ W | < s; this shows (i).Now, assume that z = s.Then If Z ⊆ W , then |Z ∩ W | < z = s, so g f Z = 0 from (i), and thus (12) evaluates to 0. On the other hand, if We define the subspace and observe first that U ⊆ span K q n .Indeed, for each T in (13) and W ′ ∈ W s , we have by Lemma 3.2 that In the latter case note that T \ W ′ ∈ E(K q n ), and the claim follows by bilinearity of .
Let Y be the orthogonal complement of U in span K q n and let Γ : span K q n → span K q n be the orthogonal projection on Y .Our main technical lemma in this paper states that Γ satisfies the assumptions of Lemma 4.1.
). Deferring the proof of Lemma 4.3, let us first compute rank Γ and conclude the proof of Theorem 1.1 assuming Lemma 4.3.
Notice that the sets Proof of Theorem 1.1.On the one hand, by Lemma 4.3 the map Γ satisfies the assumptions of Lemma 4.1.Therefore, On the other hand, Lemma 2.1 gives the same upper bound.
Proof of Lemma 4.3.We claim that m = (g First, we verify that m ∈ ker Γ = U .By Proposition 3.5 we have By Lemma 3.3 we can write each of these terms as Combining this with (4) gives 3 Let us briefly sketch the topological idea hidden behind this choice: As it can be easily deduced from the computations below, m can be also expressed as ±g (f In the terminology of simplicial complexes interpreting loosely (i) e R i as a full simplex on the vertex set R i , (ii) ∧ as a join of simplicial complexes and (iii) as an operator taking the skeleton of appropriate dimension, we gradually get the following: f J i e R i corresponds to the 0-skeleton of the simplex on R i , that is, the vertices of R i .Then (f corresponds to the join of the sets R i , that is, the complete d-partite complex on R 1 , . . ., R d .Finally, applying g to this element takes the skeleton again reducing the dimension so that the corresponding hypergraph is the required Therefore, we get where the last equality follows by Lemma 4.2(i) with z = d.Thus m ∈ U as wanted.
Next, we show that supp(m) = E(K q n [R]).As we just have shown, m ∈ U ⊆ span K q n , i.e. supp(m) ⊆ E(K q n ).Now, for T ∈ E(K q n ) we have e T , m = e T ∧ (g where for each i ∈ P the set τ i = T ∩ N i contains a single vertex.Applying Proposition 3.5, we deduce Since Turning to P ′ , denote N ′ := i∈P ′ N i \ J i .We have Therefore, g, where the second equality is due to Lemma 4.2(ii), using that there is exactly one and consequently T ∈ supp(m).

Weak saturation in the clique
In this section we prove Theorem 1.3.Let H be a q-graph where q ≥ 2 without isolated vertices.We recall the notion of a link hypergraph of a vertex v ∈ V (H): it is the (q − 1)-graph (possibly with isolated vertices) defined via L H (v) := {e \ {v} : e ∈ E(H), v ∈ e}.
The co-degree of a set W of q − 1 vertices in H is Define the minimum positive co-degree of H, in notation δ * (H), as Notice that δ * (H) ≤ δ * (L H (v)) for all v ∈ V (H), and equality holds for some v.
Proof.We apply induction on q.For q = 2 this is a well-known fact ([FGJ13], Theorem 4).Suppose now that q ≥ 3 and the statement holds for all smaller values.Let H be a q-graph and let W = {v 1 , . . ., v q−1 } be a set satisfying d H (W ) = δ * (H).Let H 1 = L H (v 1 ) be the link hypergraph of v 1 , and observe that δ * (H 1 ) = δ * (H).A weakly H-saturated q-graph on [n] is obtained as follows.Take a minimum weakly H 1 -saturated (q − 1)-graph on [n − 1] and insert n into each edge; take a union of the resulting q-graph with a minimum weakly H-saturated q-graph on [n − 1].We therefore obtain wsat(n, H) ≤ wsat(n − 1, H) + wsat(n − 1, H 1 ).
Iterating and applying the induction hypothesis, wsat(n, H) ≤ wsat(|V (H)|, H) The tensor product of two q-graphs G and J, G × J is defined having the vertex set V (G) × V (J) and the edge set (Note that every pair of edges in the original graphs produces q! edges in the product.) Proof.It suffices to prove the above statement when r 1 = • • • = r d =: r, i.e. when H = K d (r; d), as every edge creating a new copy of K d (max{r 1 , . . ., r d }; d) creates in particular a new copy of K d r1,...,r d .We apply induction on d and n.For d = 2 and any n ≥ |V (H)| the graph K [n] × K [2] misses only a matching from F 2 n , making it already H-saturated in F 2 n , as can be easily checked.Moreover, for every fixed H we can assume the statement to hold for all n less than some large C(H).
For the induction step, fix (n, d) and suppose that the statement holds for all (n ′ , d ′ ) with d ′ < d and all (n ′′ , d) with n ′′ < n.It suffices to show that O H (n d−3 ) edges can be added to G(n − 1, H) to satisfy the assertion; these edges will be as follows.
For each i ∈

By the induction hypothesis |E
As above, insert (n, i 1 ) and (n, i 2 ) into each edge of E ′ i1,i2 ; let the resulting edge set be called E i1,i2 .Finally, take all edges of F d n containing at least three vertices with n as their first coordinate, and let E 0 be this edge set; clearly 2 ) E i1,i2 is weakly H-saturated in J 2 and J 2 ∪ E 0 = F d n .Thus, G(n, H) is weakly Hsaturated in F d n as desired.This proves the induction step, and the statement of the lemma follows.We claim that G bip is weakly H-saturated in K bip via the H-saturating sequence f 1 , f ′ 1 , . . ., f k , f ′ k , where, for each ℓ ∈ [k], f ℓ = {(i ℓ , 1), (j ℓ , 2)} and f ′ ℓ = {(i ℓ , 2), (j ℓ , 1)}, and that G bip ℓ−1 ∪ {f ℓ , f ′ ℓ } = G bip ℓ for all ℓ ∈ [k] (where G bip ℓ is defined analogously, i.e., G bip ℓ = G ℓ ×K [2] ).Indeed, let (A, B) be a bipartition of V (H ℓ ) with i ℓ ∈ A and j ℓ ∈ B, and consider the analogous graph H b ℓ between A × {1} and B × {2}, i.e., for every (i, j) ∈ A × B we have {(i, 1), ( g, f e S = S⊆N e S ∧ g, f h, e S = S⊆N h, e S (e S ∧ g), f = S⊆N h, e S e S ∧ g, f = h ∧ g, f .
ti and S i ∈ Ni si , and let T := T 1 ∪ • • • ∪ T d and S := S 1 ∪ • • • ∪ S d .Then d i=1 f i = e T and d i=1 h i = e S by the definition of the exterior product ∧.If T i ⊆ S i for some i ∈ [d], then T ⊆ S and both sides of (9) vanish by Lemma 3.2.Therefore, it remains to check the case that T i ⊆ S i for every i ∈ [d].Here by Lemma 3.4 (with U = S \ T ) and Lemma 3.2 we get e T e S = sgn(S \ T, T )e S\T then T R and by Lemma 3.2 we have e T e R = 0, and consequently e T , m = 0. Hence, T / ∈ supp(m).Now assume that T ∈ E(K q n [R]), i.e., T ⊆ R. By (17) and Lemma 3.2 we have e T , m = ± g ∧ f J , e R\T (7) = ± g, f J e R\T .(18) Let P := {i ∈ [d] : T ∩ N i = ∅} and P ′ := [d] \ P .Using this notation we can write e R\T = ± i∈P e Ri\τi ∧ i∈P ′ e Ri ,
by Lemma 3.1 for every i ∈ P we have f Ji e Ri\τi ∈ 0 V .Thus f Ji e Ri\τi = e ∅ , f Ji e Ri\τi e ∅ = e ∅ ∧ f Ji , e Ri\τi e ∅ = f Ji , e Ri\τi e ∅ , and notice that f Ji , e Ri\τi = 0 because (f v ) v∈Ni is generic with respect to (e v ) v∈Ni .Plugging it into (19) yields j, 2)} ∈ E(H ℓ ) if and only if {i, j} ∈ E(H ℓ ).Note that f ℓ ∈ E(H b ℓ ) is the only edge of H b ℓ not already present in G bip ℓ−1 , therefore we can add it to the latter creating a new copy of H, namely H b ℓ .Symmetrically, taking a graph H ′b ℓ between A × {2} and B × {1} allows to add f′ ℓ .Since G ℓ = G ℓ−1 ∪ e ℓ , we have G bip ℓ−1 ∪ {f ℓ , f ′ ℓ } = G bip ℓ .Finally, note that G bip ∪ {f 1 , ..., f ′ k } = G bip k = K bip [n] .Note that K bip[n] is isomorphic to K n,n minus a perfect matching, and it is a straightforward check that this graph is H-saturated in K n,n (we can assume that |V (H)| ≤ n).We have thus shown wsat(K n,n , H) ≤ 2 wsat(n, H). and taking the limit, (1) follows readily.For the second statement, denote H = K d r1,...,r d where 1 ≤ r 1 ≤ • • • ≤ r d .Observe that the upper bound in (2) holds by Lemma 5.1, as δ * (H) = r 1 .To prove the lower bound, suppose G is weakly H-saturated in K d[n]