Colorful Subhypergraphs in Uniform Hypergraphs

There are several topological results ensuring the existence of a large complete bipartite subgraph in any properly colored graph satisfying some special topological regularity conditions. In view of $\mathbb{Z}_p$-Tucker lemma, Alishahi and Hajiabolhassan [{\it On the chromatic number of general Kneser hypergraphs, Journal of Combinatorial Theory, Series B, 2015}] introduced a lower bound for the chromatic number of Kneser hypergraphs ${\rm KG}^r({\mathcal H})$. Next, Meunier [{\it Colorful subhypergraphs in Kneser hypergraphs, The Electronic Journal of Combinatorics, 2014}] improved their result by proving that any properly colored general Kneser hypergraph ${\rm KG}^r({\mathcal H})$ contains a large colorful $r$-partite subhypergraph provided that $r$ is prime. In this paper, we give some new generalizations of $\mathbb{Z}_p$-Tucker lemma. Hence, improving Meunier's result in some aspects. Some new lower bounds for the chromatic number and local chromatic number of uniform hypergraphs are presented as well.


Background and Motivations.
In 1955, Kneser [18] posed a conjecture about the chromatic number of Kneser graphs. In 1978, Lovász [20] proved this conjecture by using algebraic topology. The Lovász's proof marked the beginning of the history of topological combinatorics. Nowadays, it is an active stream of research to study the coloring properties of graphs by using algebraic topology. There are several lower bounds for the chromatic number of graphs related to the indices of some topological spaces defined based on the structure of graphs. However, for hypergraphs, there are a few such lower bounds, see [6,11,17,19,27].
A hypergraph H is a pair (V (H), E(H)), where V (H) is a finite set, called the vertex set of H, and E(H) is a family of nonempty subsets of V (H), called the edge set of H. Throughout the paper, by a nonempty hypergraph, we mean a hypergraph with at least one edge. If any edge e ∈ E(H) has the cardinality r, then the hypergraph H is called r-uniform. For a set U ⊆ V (H), the induced subhypergraph on U , denoted H[U ], is a hypergraph with the vertex set U and the edge set {e ∈ E(H) : e ⊆ U }. Throughout the paper, by a graph, we mean a 2-uniform hypergraph. Let r ≥ 2 be a positive integer and q ≥ r be an integer. An r-uniform hypergraph H is called q-partite with parts V 1 , . . . , V q if • each edge of H intersects each part V i in at most one vertex.
If H contains all possible edges, then we call it a complete r-uniform q-partite hypergraph. Also, we say the hypergraph H is balanced if the values of |V j | for j = 1, . . . , q differ by at most one, i.e., |V i | − |V j | ≤ 1 for each i, j ∈ [q].
Let H be an r-uniform hypergraph and U 1 , . . . , U q be q pairwise disjoint subsets of V (H). The hypergraph H[U 1 , . . . , U q ] is a subhypergraph of H with the vertex set Note that H[U 1 , . . . , U q ] is an r-uniform q-partite hypergraph with parts U 1 , . . . , U q . By the symbol [n] r , we mean the family of all r-subsets of the set [n]. The hypergraph K r n = [n], [n] r is celled the complete r-uniform hypergraph with n vertices. For r = 2, we would rather use K n instead of K 2 n . The largest possible integer n such that H contains K r n as a subhypergraph is called the clique number of H, denoted ω(H).
A proper t-coloring of a hypergraph H is a map c : V (H) −→ [t] such that there is no monochromatic edge. The minimum possible such a t is called the chromatic number of H, denoted χ(H). If there is no such a t, we define the chromatic number to be infinite. Let c be a proper coloring of H and U 1 , . . . , U q be q pairwise disjoint subsets of V (H). The hypergraph H[U 1 , . . . , U q ] is said to be colorful if for each j ∈ [q], the vertices of U j get pairwise distinct colors. For a properly colored graph G, a subgraph is called multicolored if its vertices get pairwise distinct colors.
For a hypergraph H, the Kneser hypergraph KG r (H) is an r-uniform hypergraph with the vertex set E(H) and whose edges are formed by r pairwise vertex-disjoint edges of H, i.e., For any graph G, it is known that there are several hypergraphs H such that KG 2 (H) and G are isomorphic. The Kneser hypergraph KG r K k n is called the "usual" Kneser hypergraph which is denoted by KG r (n, k). Coloring properties of Kneser hypergraphs have been studied extensively in the literature. Lovász [20] (for r = 2) and Alon, Frankl and Lovász [7] determined the chromatic number of KG r (n, k). For an integer r ≥ 2, they proved that For a hypergraph H, the r-colorability defect of H, denoted cd r (H), is the minimum number of vertices which should be removed such that the induced hypergraph on the remaining vertices is r-colorable, i.e., cd r (H) = min {|U | : For a hypergraph H, Dol'nikov [11] (for r = 2) and Kříž [19] proved taht which is a generalization of the results by Lovász [20] and Alon, Frankl and Lovász [7]. For a positive integer r, let Z r = {ω, ω 2 . . . , ω r } be a cyclic group of order r with generator ω. Consider a vector X = (x 1 , x 2 , . . . , x n ) ∈ (Z r ∪ {0}) n . An alternating subsequence of X is a sequence x i 1 , x i 2 , . . . , x im of nonzero terms of X such that i 1 < · · · < i m and x i j = x i j+1 for each j ∈ [m − 1]. We denote by alt(x) the maximum possible length of an alternating subsequence of X. For a vector X = (x 1 , x 2 , . . . , x n ) ∈ (Z r ∪ {0}) n and for an ǫ ∈ Z p , set X ǫ = {i ∈ [n] : x i = ǫ}. Note that, by abuse of notation, we can write X = (X ǫ ) ǫ∈Zr . For two vectors X, Y ∈ (Z r ∪ {0}) n , by X ⊆ Y , we mean X ǫ ⊆ Y ǫ for each ǫ ∈ Z r .
For a hypergraph H and a bijection σ : Also, let where the minimum is taken over all bijection σ : [n] −→ V (H). One can readily check that for any hypergraph H, |V (H)| − alt r (H) ≥ cd r (H) and the inequality is often strict, see [6]. Alishahi and Hajiabolhassan [6] improved Dol'nikov-Kříž result by proving that for any hypergraph H and for any integer r ≥ 2, the quantity |V (H)|−altr(H) r−1 is a lower bound for the chromatic number of KG r (H). Using this lower bound, the chromatic number of some families of graphs and hypergraphs are computed, see [1,2,3,4,6,14]. There are some other lower bounds for the chromatic number of graphs which are better than the former discussed lower bounds. They are based on some topological indices of some topological spaces connected to the structure of graphs. In spite of these lower bounds being better, they are not combinatorial and most of the times they are difficult to compute.
The existence of large colorful bipartite subgraphs in a properly colored graph has been extensively studied in the literature [6,8,9,26,27,28]. To be more specific, there are several theorems ensuring the existence of a colorful bipartite subgraph in any properly colored graph such that the bipartite subgraph has a specific number of vertices related to some topological parameters connected to the graph. Simonyi and Tardos [28] improved Dol'nikov's lower bound and proved that in any proper coloring of a Kneser graph KG 2 (H), there is a multicolored complete bipartite graph K cd 2 (H) 2 , cd 2 (H) 2 such that the cd 2 (H) different colors occur alternating on the two parts of the bipartite graph with respect to their natural order. By a combinatorial proof, Alishahi and Hajiabolhassan [6] improved this result. They proved that the the result remains true if we replace cd 2 (H) by n − alt 2 (H). Also, a stronger result is proved by Simonyi, Tardif, and Zsbán [26].
Theorem A. (Zig-zag Theorem [26]). Let G be a nonempty graph which is properly colored with arbitrary number of colors. Then G contains a multicolored complete bipartite subgraph K ⌈ t 2 ⌉,⌊ t 2 ⌋ , where Xind(Hom(K 2 , G)) + 2 = t. Moreover, colors appear alternating on the two sides of the bipartite subgraph with respect to their natural ordering.
The quantity Xind(Hom(K 2 , G)) is the cross-index of hom-complex Hom(K 2 , G) which will be defined in Subsection 2.2. We should mention that there are some other weaker similar results in terms of some other topological parameters, see [27,28].
Note that prior mentioned results concern the existence of colorful bipartite subgraphs in properly colored graphs (2-uniform hypergraphs). In 2014, Meunier [23] found the first colorful type result for the uniform hypergraphs. He proved that for any prime number p, any properly colored Kneser hypergraph KG p (H) must contain a colorful balanced complete p-uniform p-partite subhypergraph with a specific number of vertices, see Theorem C.

Main
Results. For a given graph G, there are several complexes defined based on the structure of G. For instance, the box-complex of G, denoted B 0 (G), and the hom-complex of G, denoted Hom(K 2 , G), see [21,26,27]. Also, there are some lower bounds for the chromatic number of graphs related to some indices of these complexes [26,27]. In this paper, we naturally generalize the definitions of box-complex and hom-complex of graphs to uniform hypergraphs. Also, the definition of Z p -cross-index of Z p -posets will be introduced. Using these complexes, as a first main result of this paper, we generalize Meunier's result [23] (Theorem C) to the following theorem. (i) There is some colorful balanced complete r-uniform p-partite subhypergraph in H with ind Zp (B 0 (H, Z p )) + 1 vertices. In particular, (ii) If p ≤ ω(H), then there is some colorful balanced complete r-uniform p-partite subhypergraph in H with Xind Zp (Hom(K r p , H)) + p vertices. In particular, Quantities ind Zp (B 0 (H, Z p )) and Xind Zp (Hom(K r p , H)) appearing in the statement of Theorem 1 are respectively the Z p -index and Z p -cross-index of the Z p -box-complex B 0 (H, Z p ) and Z p -homcomplex Hom(K r p , H) which will be defined in Subsection 2.2. Using these complexes, we introduce some new lower bounds for the chromatic number of uniform hypergraphs. In view of Theorem 1, next theorem provides a hierarchy of lower bounds for the chromatic number of r-uniform hypergraphs.
Theorem 2. Let r ≥ 2 be a positive integer and p ≥ r be a prime number. For any r-uniform hypergraph H, we have the following inequalities.
In view of Theorem 2, Theorem 1 is a common extension of Theorem A and Theorem C. Furthermore, for r = 2, Theorem 1 implies the next corollary which also is a generalization of Theorem A. Corollary 1. Let p be a prime number and let G be a nonempty graph which is properly colored with arbitrary number of colors. Then there is a multicolored complete p-partite subgraph K n 1 ,n 2 ,...,np of G such that In view of the prior mentioned results, the following question naturally arises. Question 1. Do Theorem 1 and Theorem 2 remain true for non-prime p?

Applications to Local Chromatic Number of Uniform Hypergraphs. For a graph G and a vertex
The local chromatic number of G, denoted χ l (G), is defined in [12] as follows: where the minimum in taken over all proper coloring c of G. Note that Theorem A gives the following lower bound for the local chromatic number of a nonempty graph G: Note that for a Kneser hypergraph KG 2 (H), by using Simonyi and Tardos colorful result [28] or the extension given by Alishahi and Hajiabolhassan [6], there are two similar lower bounds for χ l (KG 2 (H)) which respectively used cd 2 (H) and |V (H)|−alt 2 (H) instead of Xind(Hom(K 2 , G))+2. However, as it is stated in Theorem 2, the lower bound in terms of Xind(Hom(K 2 , G)) + 2 is better than these two last mentioned lower bounds. Using Corollary 1, we have the following lower bound for the local chromatic number of graphs.
Corollary 2. Let G be a nonempty graph and p be a prime number. Then Note that if we set p = 2, then previous theorem implies Simonyi and Tardos lower bound for the local chromatic number. Note that, in general, this lower bound might be better than Simonyi and Tardos lower bound. To see this, let k ≥ 2 be a fixed integer. Consider the Kneser graph KG 2 (n, k) and let p = p(n) be a prime number such that p = O(ln n). By Theorem 2, for n ≥ pk, . Note that the lower bound for χ l (KG 2 (n, k)) coming form Inequality 1 is while, in view of Corollary 2, we have which is better than the quantity in Equation 2 if n is sufficiently large. However, since the induced subgraph on the neighbors of any vertex of KG(n, k) is isomorphic to KG(n − k, k), we have Corollary 3. Let F be a hypergraph and α(F) be its independence number. Then for any prime number p, we have Proof. In view of Theorem 2, we have Now, Corollary 2 implies the assertion.
Meunier [23] naturally generalized the definition of local chromatic number of graphs to uniform hypergraphs as follows. Let H be a uniform hypergraph. For a set X ⊆ V (H), the closed neighborhood of X is the set X ∪ N (X), where For a uniform hypergraph H, the local chromatic number of H is defined as follows: where the minimum is taken over all proper coloring c of H.
Meunier [23], by using his colorful theorem (Theorem C), generalized Simonyi and Tardos lower bound [28] for the local chromatic number of Kneser graphs to the local chromatic number of Kneser hypergraphs. He proved: for any hypergraph H and any prime number p. In what follows, we generalize this result.
Theorem 3. Let H be an r-uniform hypergraph with at least one edge and p be a prime number, where r ≤ p ≤ ω(H). Let t = Xind Zp (Hom(K r p , H))+p. If t = ap+b, where a and b are nonnegative integers and 0 ≤ b ≤ p − 1, then Proof. Let c be an arbitrary proper coloring of H and let H[U 1 , . . . , U p ] be the colorful balanced complete r-uniform p-partite subhypergraph of H whose existence is ensured by Theorem 1. Note that b numbers of U i 's, say U 1 , . . . , U b , have the cardinality ⌈ t r ⌉ while the others have the cardinality ⌊ t r ⌋ ≥ 1. Consider U 1 , . . . , U p−r+1 . Two different cases will be distinguished.
Therefore, since any color is appeared in at most r − 1 number of U i 's, we have and consequently, which completes the proof in Case 1.
which completes the proof in Case 2.

Corollary 4.
Let H be a p-uniform hypergraph with at least one edge, where p is a prime number. Then Proof. Since H has at least one edge, we have ω(H) ≥ p. Therefore, in view of Theorem 3, we have the assertion.
Note that if H = KG p (F), then, in view of Theorem 2, we have Xind Zp (Hom(K p p , H)) + p ≥ |V (F)| − alt p (F). This implies that the previous corollary is a generalization of Meunier's lower bound for the local chromatic number of KG p (F) 1.4. Plan. Section 2 contains some backgrounds and essential definitions used elsewhere in the paper. In Section 3, we present some new topological tools which help us for the proofs of main results. Section 4 is devoted to the proofs of Theorem 1 and Theorem 2.

Topological Indices and Lower
Bound for Chromatic Number. We assume basic knowledge in combinatorial algebraic topology. Here, we are going to bring a brief review of some essential notations and definitions which will be needed throughout the paper. For more, one can see the book written by Matoušek [21]. Also, the definitions of box-complex, hom-complex, and cross-index will be generalized to Z p -box-complex, Z p -hom-complex, and Z p -cross-index, respectively.
Let G be a finite nontrivial group which acts on a topological space X. We call X a topological G-space if for each g ∈ G, the map g : X −→ X which x → g · x is continuous. A free topological G-space X is a topological G-space such that G acts on it freely, i.e., for each g ∈ G \ {e}, the map g : X −→ X has no fixed point. For two topological G-spaces X and Y , a continuous map Simplicial complexes provide a bridge between combinatorics and topology. A simplicial complex can be viewed as a combinatorial object, called abstract simplicial complex, or as a topological space, called geometric simplicial complex. Here, we just remind the definition of an abstract simplicial complex. However, we assume that the reader is familiar with the concept of how an abstract simplicial complex and its geometric realization are connected to each other. A simplicial complex is a pair (V, K), where V is a finite set and K is a family of subsets of V such that if F ∈ K and F ′ ⊆ F , then F ′ ∈ K. Any set in K is called a simplex. Since we may assume that V = F ∈K F , we can write K instead of (V, K). The dimension of K is defined as follows: The geometric realization of K is denoted by ||K||. For two simplicial complexes C and K, by a simplicial map f : C −→ K, we mean a map from V (C) to V (K) such that the image of any simplex of C is a simplex of K. For a nontrivial finite group G, a simplicial G-complex K is a simplicial complex with a G-action on its vertices such that each g ∈ G induces a simplicial map from K to K, that is the map which maps v to g · v for each v ∈ V (K). If for each g ∈ G \ {e}, there is no fixed simplex under the simplicial map made by g, then K is called a free simplicial G-complex. For a simplicial G-complex K, if we take the affine extension, then K is free if and only if ||K|| is free. For two simplicial G-complexes C and K, a simplicial map f : For an integer n ≥ 0 and a nontrivial finite group G, E n G space is a free (n − 1)-connected n-dimensional simplicial G-complexes. A concrete example of an E n G space is the (n + 1)-fold join G * (n+1) . As a topological space G * (n+1) is a (n + 1)-fold join of an (n + 1)-point discrete space. This is known that for any two E n G space X and Y , there is a G-map from X to Y .
For a G-space X, define ind G (X) = min{n : X G −→ E n G}.
Note that here E n G can be any E n G, since there is a G-map between any two E n G spaces, see [21]. Also, for a simplicial complex K, by ind G (K), we mean ind G (||K||). Throughout the paper, for G = Z 2 , we would rather use ind(−) instead of ind Z 2 (−).
Properties of the G-index. [21] Let G be a finite nontrivial group.
2.2. Z p -Box-Complex, Z p -Poset, and Z p -Hom-Complex. In this subsection, for any r-uniform hypergraph H, we are going to define two objects; Z p -box-complex of H and Z p -hom-complex of H which the first one is a simplicial Z p -complex and the second one is a Z p -poset. Moreover, for any Z p -poset P , we assign a combinatorial index to P called the cross-index of P . • the hypergraph H[U 1 , U 2 , . . . , U p ] is a complete r-uniform p-partite hypergraph. Note that some of U i 's might be empty. In fact, if U 1 , . . . , U p are pairwise disjoint subsets of V (H) and the number of nonempty U i 's is less than r, then H[U 1 , U 2 , . . . , U p ] is a complete runiform p-partite hypergraph and thus {ω 1 } × U 1 ∪ · · · ∪ {ω p } × U p ∈ B 0 (H, Z p ). For each ǫ ∈ Z p and each (ǫ ′ , v) ∈ V (B 0 (H, Z p )), define ǫ · (ǫ ′ , v) = (ǫ · ǫ ′ , v). One can see that this action makes B 0 (H, Z p ) a free simplicial Z p -complex. It should be mentioned that the Z 2 -box-complex B 0 (H, Z 2 ) is extensively studied in the literature, see [27,28]. In the literature, for a graph G, the simplicial complex B 0 (G, Z 2 ) is shown by B 0 (G). This simplicial complex is used to introduce some lower bounds for the chromatic number of a given graph G, see [27]. In particular, we have the following inequalities where F is any hypergraph such that KG 2 (F) and G are isomorphic, see [2,6,27].
Z p -Poset. A partially ordered set, or simply a poset, is defined as an ordered pair P = (V (P ), ), where V (P ) is a set called the ground set of P and is a partial order on V (P ). For two posets P and Q, by an order-preserving map φ : P −→ Q, we mean a map φ from V (P ) to V (Q) such that for each u, v ∈ V (P ), if u v, then φ(u) φ(v). A poset P is called a Z p -poset, if Z p acts on V (P ) and furthermore, for each ǫ ∈ Z p , the map ǫ : V (P ) −→ V (P ) which v → ǫ · v is an automorphism of P (order preserving bijective map). If for each ǫ ∈ Z p \ {e}, this map has no fixed point, then P is called a free Z p -poset. For two Z p -poset P and Q, by an order-preserving Z p -map φ : P −→ Q, we mean an order-preserving map from V (P ) to V (Q) such that for each v ∈ V (P ) and ǫ ∈ Z p , we If there exists such a map, we write P Zp −→ Q. For a nonnegative integer n and a prime number p, let Q n,p be a free Z p -poset with ground set Z p × [n + 1] such that for any two members (ǫ, i), (ǫ ′ , j) ∈ Q n,p , (ǫ, i) < Qn,p (ǫ ′ , j) if i < j. Clearly, Q n,p is a free Z p -poset with the action ǫ · (ǫ ′ , j) = (ǫ · ǫ ′ , j) for each ǫ ∈ Z p and (ǫ ′ , j) ∈ Q n,p . For a Z p -poset P , the Z p -cross-index of P , denoted Xind Zp (P ), is the least integer n such that there is a Z p -map from P to Q n,p . Throughout the paper, for p = 2, we speak about Xind(−) rather than Xind Z 2 (−). It should be mentioned that Xind(−) is first defined in [26].
Let P be a poset. We can define an order complex ∆P with the vertex set same as the ground set of P and simplex set consisting of all chains in P . One can see that if P is a free Z p -poset, then ∆P is a free simplicial Z p -complex. Moreover, any order-preserving Z p -map φ : P −→ Q can be lifted to a simplicial Z p -map from ∆P to ∆Q. Clearly, there is a simplicial Z p -map from ∆Q n,p to Z * (n+1) p (identity map). Therefore, if Xind Zp (P ) = n, then we have a simplicial Z p -map from ∆P to Z * (n+1) p . This implies that Xind Zp (P ) ≥ ind Zp (∆P ). Throughout the paper, for each (ǫ, j) ∈ Q n,p , when we speak about the sign of (ǫ, j) and the absolute value of (ǫ, j), we mean ǫ and j, respectively.
Z p -Hom-Complex. Let H be an r-uniform hypergraph. Also, let p ≥ r be a prime number. The Z p -hom-complex Hom(K r p , H) is a free Z p -poset with the ground set consisting of all ordered p-tuples (U 1 , · · · , U p ), where U i 's are nonempty pairwise disjoint subsets of V and H[U 1 , . . . , U p ] is a complete r-uniform p-partite hypergraph. For two p-tuples (U 1 , · · · , U p ) and (U ′ 1 , · · · , U ′ p ) in Hom(K r p , H), we define (U 1 , · · · , U p ) . Also, for each ω i ∈ Z p = {ω 1 , . . . , ω p }, let ω i · (U 1 , · · · , U p ) = (U 1+i , · · · , U p+i ), where U j = U j−p for j > p. Clearly, this action is a free Z p -action on Hom(K r p , H). Consequently, Hom(K r p , H) is a free Z pposet with this Z p -action.
For a nonempty graph G and for p = 2, it is proved [2,6,26,27] that where F is any hypergraph such that KG 2 (F) and G are isomorphic.

Notations and Tools
For a simplicial complex K, by sd K, we mean the first barycentric subdivision of K. It is the simplicial complex whose vertex set is the set of nonempty simplices of K and whose simplices are the collections of simplices of K which are pairwise comparable by inclusion. Throughout the paper, by σ r−1 t−1 , we mean the (t − 1)-dimensional simplicial complex with vertex set Z r containing all t-subsets of Z r as its maximal simplices. The join of two simplicial complexes C and K, denoted C * K, is a simplicial complex with the vertex set V (C) V (K) and such that the set of its simplices is {F 1 F 2 : F 1 ∈ C and F 2 ∈ K}. Clearly, we can see Z r as a 0-dimensional simplicial complex. Note that the vertex set of simplicial complex sd Z * α r can be identified with (Z r ∪ {0}) α \ {0} and the vertex set of (σ r−1 t−1 ) * n is the set of all pairs (ǫ, i), where ǫ ∈ Z r and i ∈ [n].
3.1. Z p -Tucker-Ky Fan lemma. The famous Borsuk-Ulam theorem has many generalizations which have been extensively used in investigating graph coloring properties. Some of these interesting generalizations are Tucker lemma [29], Z p -Tucker Lemma [30], and Tucker-Ky Fan [13]. For more details about the Borsuk-Ulam theorem and its generalizations, we refer the reader to [21]. Actually, Tucker lemma is a combinatorial counterpart of Borsuk-Ulam theorem. There are several interesting and surprising applications of Tucker Lemma in combinatorics, including a combinatorial proof of Lovász-Kneser theorem by Matoušek [22]. • for any X ∈ {−1, 0, +1} n \ {0}, we have λ(−X) = −λ(X) (a Z 2 -equivariant map), • no two signed vectors X and Y are such that X ⊆ Y and λ(X) = −λ(Y ).
Then, we have m ≥ n.
Another interesting generalization of the Borsuk-Ulam theorem is Ky Fan's lemma [13]. This generalization ensures that with the same assumptions as in Lemma A, there is odd number of chains X 1 ⊆ X 2 ⊆ · · · ⊆ X n such that where 1 ≤ c 1 < · · · < c n ≤ m. Ky Fan's lemma has been used in several articles to study some coloring properties of graphs, see [5,9,15]. There are also some other generalizations of Tucker Lemma. A Z p version of Tucker Lemma, called Z p -Tucker Lemma, is proved by Ziegler [30] and extended by Meunier [25]. In next subsection, we present a Z p version of Ky Fan's lemma which is called Z p -Tucker-Ky Fan lemma.
be a Z p -equivariant map satisfying the following conditions.
Proof. Note that the map λ can be considered as a simplicial Z p -map from sd Z * n p to (Z * α p ) * ((σ p−1 p−2 ) * (m−α) ). Let K = Im(λ). Note that each simplex in K can be represented in a unique form σ ∪ τ such that σ ∈ Z * α p and τ ∈ (σ p−1 p−2 ) * m−α . In view of definition of the function l(−) and the properties which λ satisfies in, to prove the assertion, it suffices to show that there is a simplex σ ∪ τ ∈ K such that l(τ ) ≥ n − α. For a contradiction, suppose that for each σ ∪ τ ∈ K, we have l(τ ) ≤ n − α − 1.
Define the map Γ : sd K −→ Z * (n−1) p such that for each vertex σ ∪ τ ∈ V (sd K), • if τ = ∅, then Γ(σ ∪ τ ) = (ǫ, j), where j is the maximum possible value such that (ǫ, j) ∈ σ. Note that since σ ∈ Z * α p , there is only one ǫ ∈ Z p for which the maximum is attained. Therefore, in this case, the function Γ is well-defined.
a contradiction. Therefore, Γ is a simplicial Z p -map from sd K to Z * (n−1) p . Naturally, λ can be lifted to a simplicial Z p -mapλ : sd 2 Z * n p −→ sd K. Thus Γ •λ is a simplicial Z p -map from sd 2 Z * n p to Z * (n−1) p . In view of Dold's theorem [10,21], the dimension of Z * (n−1) p should be strictly larger than the connectivity of sd 2 Z * n p , that is n − 2 > n − 2, which is not possible.
Lemma 1 provides a short simple proof of Meunier's colorful result for Kneser hypergraphs (next Theorem) as follows.
Theorem C. [23] Let H be a hypergraph and let p be a prime number. Then any proper coloring c : V (KG p (H)) −→ [C] (C arbitrary) must contain a colorful balanced complete p-uniform p-partite hypergraph with |V (H)| − alt p (H) vertices.
• If alt(X) ≤ alt p (H, π), then let λ 1 (X) be the first nonzero coordinate of X and λ 2 (X) = alt(X). • If alt(X) ≥ alt p (H, π) + 1, then in view of the definition of alt p (H, π), there is some ǫ ∈ Z p such that E(π(X ǫ )) = ∅. Define c(X) = max {c(e) : ∃ǫ ∈ Z p such that e ⊆ π(X ǫ )} and λ 2 (X) = alt p (H, π) + c(X). Choose ǫ ∈ Z p such that there is at least one edge e ∈ π(X ǫ ) with c(X) = c(e) and such that X ǫ is the maximum one having this property. By the maximum, we mean the maximum according to the total ordering . It is clear that ǫ is defined uniquely. Now, let λ 1 (X) = ǫ. One can check that λ satisfies the conditions of Lemma 1. Consider the chain Z 1 ⊂ Z 2 ⊂ · · · ⊂ Z n−altp(H,π) whose existence is ensured by Lemma 1. Note that for each i ∈ [n−alt p (H, π)], we have λ 2 (Z i ) > alt p (H, π). Consequently, λ 2 (Z i ) = alt p (H, π) + c(Z i ). Let λ(Z i ) = (ǫ i , j i ). Note that for each i, there is at least one edge e i,ǫ i ⊆ π(Z ǫ i i ) ⊆ π(Z ǫ i n−altp(H,π) ) such that c(e i,ǫ i ) = j i − alt p (H, π). For each ǫ ∈ Z p , define U ǫ = {e i,ǫ i : ǫ i = ǫ}. We have the following three properties for U ǫ 's.
The proof of next lemma is similar to the proof of Lemma 1.

Lemma 2.
Let C be a free simplicial Z p -complex such that ind Zp (C) ≥ t and let λ : C −→ (σ p−1 p−2 ) * m be a simplicial Z p -map. Then there is at least one t-dimensional simplex σ ∈ C such that τ = λ(σ) is a t-dimensional simplex and for each ǫ ∈ Z p , we have ⌊ t+1 p ⌋ ≤ |τ ǫ | ≤ ⌈ t+1 p ⌉. Proof. For simplicity of notation, let K = Im(λ). Clearly, to prove the assertion, it is enough to show that there is a t-dimensional simplex τ ∈ K such that l(τ ) ≥ t. Suppose, contrary to the assertion, that there is no such a t-dimensional simplex. Therefore, for each simplex τ of K, we have l(τ ) ≤ t. For each vertex τ ∈ V (sd K), set h(τ ) = min ǫ∈Zp |τ ǫ |.
Let Γ : sd K −→ Z * t p be a map such that for each vertex τ of sd K, Γ(τ ) is defined as follows.
Proof of Theorem 2. It is simple to prove that |V (F)| − alt p (F) ≥ cd p (F) for any hypergraph F. Therefore, the proof follows by Proposition 2 and Proposition 3.