1. Introduction

This paper studies the minimum number of non-monochromatic simplices that can appear in a Sperner labeling of a triangulation of a simplex. Here, a simplex is called non-monochromatic if at least two of its vertices receive different labels. This question is related to hypergraph labeling problems and to several topics in combinatorial topology and discrete optimisation; for related work, see [1,2].

A main motivation comes from the hypergraph labeling problem, where one seeks a cut minimizing the sum of assignment costs of the vertices and the weights of the cut hyperedges. Lower bounds on the number of non-monochromatic simplices provide structural information for these problems; see [3–5] for background.

An open problem discussed in [6] is finding a lower bound for the number of non-monochromatic simplices in Sperner labelings of simplicial subdivisions (triangulations) of the simplex , where denotes the set of vectors in with non-negative entries that sum to q. Specifically, on the final page of [6], Mirzakhani and Vondrák proposed the following open question and conjecture:

“For a Sperner-admissible labeling of a regular simplicial subdivision [...], what is the minimum possible number of non-monochromatic cells? [...]. We conjecture that for a fixed and as , the number of cells containing at least j colors is on the order of .”

This leads to the regular triangulation setting considered in this paper. Let For a Sperner labeling of the vertices of a triangulation of , let denote the minimum possible number of non-monochromatic simplices in a triangulation of . On the final page of [7], Mirzakhani and Vondrák conjectured that, for fixed and as , the number of cells containing at least j colors is of order . In the case j = 2, this predicts growth of order for the quantity studied here.

In this paper, we study this question for the regular triangulation , a symmetric triangulation of whose vertices are the lattice points of the simplex. Our approach is to characterise in graph-theoretic terms and then relate it to the simplex-lattice hypergraph considered in [7]. This allows us to prove a lower bound and an upper bound of the same order for fixed k, and therefore to determine the order of growth of up to the multiplicative constant. In particular, this order of growth is , which matches the asymptotic lower bound of the conjecture of [6] for j = 2. We also show that the bounds are exact in the initial cases q = 1 or k = 2, and we obtain the exact value of .

We establish the following main results:

Theorem 1.1 The minimum number of non-monochromatic simplices for a Sperner labeling of the vertices of the regular triangulation of satisfies

(1.1)

This bound is proved to be tight for the initial cases q = 1 or k = 2. For q = 2 we give a better lower bound with an improvement of the multiplicative constant.

Note that this lower bound is on the order of as desired. Next, we give an upper bound for the minimum number of non-monochromatic simplices, which is , with the multiplicative constant depending on the dimension k. Concretely,

Theorem 1.2 The minimum number of non-monochromatic simplices for a Sperner labeling of the vertices of the regular triangulation of satisfies

(1.2)

This paper is organised as follows. Section 2 introduces definitions and notation used throughout. In Section 3, we present and characterise the regular triangulation . Section 4 establishes a lower bound on the minimum number of non-monochromatic simplices within this triangulation. In Section 5, we determine an upper bound. Finally, in Section 6, we give some conclusions and future directions of work.

2. Basic concepts and notation

In this section, we fix notation and recall some definitions.

3. The regular triangulation

We will employ the simplicial subdivision of defined in [7] (see also [8,9]), which we refer to as the regular triangulation . The purpose of this section is to obtain a concrete description of in terms of adjacency among lattice points. To do so, we proceed in three steps. First, we introduce an auxiliary region and a triangulation of this region. Next, we encode the simplices of through the graph and then replace it with the more explicit graph , proving that both descriptions determine the same family of simplices, namely

Finally, we transport this description from to the discrete simplex by means of the map , obtaining the graph on . This leads to Corollary 3.15, which characterises the regular triangulation as the family of convex hulls of k pairwise adjacent vertices in . Before proceeding, we introduce the following preliminary definitions.

Definition 3.1 Let with . The simplex is given by

(3.1)

We denote by the set of integer lattice points contained in , that is,

Definition 3.2 Let . A permutation is said to be consistent with w if

(3.2)

Definition 3.3 Let be a permutation consistent with . We define the simplex

(3.3)

Definition 3.4 The triangulation of the region is the collection of all simplices arising from pairs where is a permutation consistent with , i.e.,

(3.4)

Definition 3.5 The regular triangulation of the discrete simplex is the collection of simplices

(3.5)

where is the mapping given by

(3.6)

We begin with the auxiliary triangulation of and describe its simplices by means of an associated graph.

Definition 3.6 The graph is the pair , where:

• The vertex set is .
• The edge set consists of pairs such that there exist that satisfy

Definition 3.7 We define the collection as the set of convex hulls of k vertices in that are pairwise adjacent in the previous graph, that is,

(3.7)

We now establish the relationship between the sets and defined in (3.4), (3.7), respectively.

Proposition 3.8 It is satisfied that .

Proof. Let be a simplex. By construction, its k vertices belong to and lie within , therefore are pairwise adjacent in and then

which shows that .

Conversely, suppose there exist k pairwise adjacent vertices such that

Since the union of all simplices in covers , i.e.,

(3.8)

an edge of the convex hull must intersect the interior of some simplex . Since this edge is an intersection of simplices of , then contains an interior point of for some , a contradiction because the intersection of simplices of a triangulation is a common face. Therefore, our assumption must be false, and it follows that and then as desired. □

Proposition 3.8 shows that is a triangulation of . We now introduce a more explicit graph-theoretic description of this triangulation.

Definition 3.9 The graph is the pair where

• The vertex set is .
• For distinct vertices , the edge belongs to if either (3.9)
  or (3.10)

Definition 3.10 The collection is the set of convex hulls of k vertices in that are pairwise adjacent in the graph . That is,

(3.11)

Proposition 3.11 It is satisfied that .

Proof. Recall that the graphs and were introduced in Definition 3.6 and Definition 3.9, respectively. Since , it suffices to prove that . Assume that , then for some and then

Since for all , it follows that there exists an index i such that

(3.12)

Similarly, for w², there exists an index j such that

(3.13)

Assume without loss of generality that j > i. Using (3.12) and (3.13), we obtain:

• for , (3.14)
• for , (3.15)
• for , (3.16)

Therefore, by (3.14), (3.15), and (3.16), we have . □

Now we show that by induction on k. For k = 2, if , then we can assume without loss of generality that . The permutation is trivially admissible and satisfies

so and then .

Assume that and consider and . We can assume without loss of generality that for . We have two cases.

Case 1. There exists an index i such that and .

We consider

These satisfy that

so .

Since corresponds to , for some , and by assumption, it follows that , for all . Consequently, , and by the induction hypothesis, . Therefore, there exists and a permutation such that . We also have that and then , so and the simplices of the triangulation of are the convex hulls of vertices pairwise adjacent in .

The map

defines a bijection between and , where the inverse is given by

Since f just duplicates a coordinate and f^-1 removes the repeated coordinate, it follows that edges of are mapped to edges of whose vertices lie in

Conversely, edges of with vertices in are mapped back to edges of by f^-1. Therefore, is isomorphic to the induced subgraph

In this way, the set of convex hulls of vertices of pairwise adjacent in is a triangulation of

we call it T, and share a simplex of T, since share a simplex of .

Moreover, for each containing vertices of

the intersection

is a simplex of

Since these vertices are pairwise adjacent in , it follows that

We denote these simplices by .

We have

Otherwise,

is a nonempty open subset of , so it has dimension . But since

we would have that

is the union of a finite number of sets with dimension , a contradiction.

We also have that

is either empty or a common face for , , since is either empty or a common face for , .

It follows that

is a triangulation of

included in T, and then

This implies that , for some , and then . Therefore, in this case.

Case 2. Suppose there does not exist an index i such that

We can assume that , , for some j, and for , while for .

We define the following permutation of : . Let us see that is consistent with w² if either or and .

Indeed, is strictly increasing except for (maybe) j,j + 1, so if , then is consistent with w². If and (that is to say, if for ), then and is strictly increasing, so is consistent with w².

Moreover, we have

and

Hence, it follows that and then in this case.

If and , then by the assumption.

Since for every t, the former implies that and , so , and then the permutation

satisfies that if , then , except for the cases , i = j + 1. So, if and , then is admissible.

In the case , we obtain

As a result, , a contradiction.

Similarly, if , then

so , a contradiction. Therefore, is admissible, with

and

From which it follows that and then , so as desired. □

Corollary 3.12 The triangulations , , and are equal.

Proof. Recall that , , and are defined by (3.4), (3.7), and (3.11), respectively. By Proposition 3.11, the graphs and are isomorphic, then the sets of k vertices that are pairwise adjacent in and coincide. Thus, the collections of simplices defined as the convex hulls of such sets are equal, that is, .

On the other hand, by Proposition 3.8, we have .

Combining these equalities, it follows that

Definition 3.13 The graph is the pair , where □

• The vertex set is .
• For distinct vertices , the edge belongs to if (3.17)
  and the number of entries equal to 1 and are equal and when reading the nonzero entries from left to right, the signs alternate.

We now establish the relationship between the graphs and introduced in Definitions 3.9 and 3.13, respectively.

Proposition 3.14 is isomorphic to .

Proof. The mapping , where is given by (3.6), is bijective, with inverse

(3.18)

If , we may assume without loss of generality that for all . Let i₀ denote the smallest index such that .

If i₀ > 1, then for all i < i₀, we have

(3.19)

and

(3.20)

If i₀ = 1, the relation simplifies to

(3.21)

Next, let i₁ be the smallest index greater than i₀ such that , if such an index exists. If , then for all we have

(3.22)

and at i₁ we have

(3.23)

Using (3.19)–(3.23) (or (3.21) when i₀ = 1), and iterating the same argument with i₂, i₃, etc, we conclude that the nonzero entries of alternate between 1 and . Moreover, we have the same number of 1’s and ’s since

Therefore, the numbers of 1’s and ’s coincide, and hence .

Conversely, if , then we can assume without loss of generality that the lowest index i such that , denoted by i₁, satisfies . If , j = 1, ..., 2t, is the ordered list of indices such that , then we have for odd j and for even j.

Consequently, if i₁ > 1, then, for i < i₁,

(3.24)

where

(3.25)

Moreover, for the indices i such that (if any), we obtain

(3.26)

with

(3.27)

Using (3.24)–(3.27) and iterating the same argument with i₃, i₄, etc., we obtain that

and then by (3.18), .

So is isomorphic to via the map (3.6), as desired. □

Corollary 3.15 The regular triangulation of can be described as

(3.28)

Proof. By Definition 3.5 and the definition of in (3.6),

(3.29)

By Corollary 3.12, we have , so the simplices of are precisely the convex hulls , where are pairwise adjacent in . Since defines a graph isomorphism between and by Proposition 3.14, the image of each such simplex under is the convex hull of k vertices in that are pairwise adjacent in . Therefore, (3.29) is equivalent to (3.28), which proves the result. □

4. The lower bound

In order to establish a lower bound for the number of non-monochromatic simplices, we need a preliminary definition of a hypergraph included in [6].

Definition 4.1 The Simplex-Lattice Hypergraph is a pair , where:

• The vertex set is .
• The hyperedge set is (4.1)

Definition 4.2. Let T be a triangulation of whose vertices are labeled with a Sperner labeling . A simplex in T is said to be monochromatic if all its vertices are assigned the same label by c; otherwise, the simplex is called non-monochromatic.

We denote by the minimum number of non-monochromatic simplices that appear in any Sperner labeling of the regular triangulation of , and we shall prove the bound given in (1.1).

Proof. The simplices associated with the hyperedges in (4.1) are simplices in the regular triangulation . Indeed, observe that for any two vertices

(4.2)

the difference is

(4.3)

which satisfies (3.17). Then, all vertices in each hyperedge of form a simplex in .

Therefore, the number of non-monochromatic simplices in is at least the number of non-monochromatic hyperedges in . According to Proposition 2.1 of [6], this number is bounded below by . □

Let us see an example for which the lower bound of Theorem 1.1 is not tight.

Example 4.3 For q = 2, any Sperner labeling of the vertices of the regular triangulation has at most one monochromatic simplex.

Indeed, if every monochromatic simplex has label k (for instance), then the monochromatic simplices have the vertices either in or in . But for q = 2, there is just one simplex with this property: the simplex with vertices in and a vertex in , namely .

If there exist monochromatic simplices with labels 1 and k (for instance), then we would have only one simplex with label 1 and only one simplex with label k as we have seen before, so the vertices in would have label 1 and the vertices in would have label k, so we obtain vertices in

with two labels, a contradiction.

Therefore, the number of non-monochromatic simplices is at least , and then . This bound improves the lower bound (1.1), for k > 2.

5. An upper bound of m(k,q)

We now establish an upper bound for .

Proof of Theorem 1.2 Consider the first choice labeling (Definition 2.4). In this labeling, a simplex is non-monochromatic if and only if it contains at least one vertex with first coordinate x₁ = 0. These are all the simplices of the regular triangulation except for the simplices such that their vertices have first coordinate .

Since the simplices of the regular triangulation with a vertex in x₁ = 0 are included in , then the simplices of such that their vertices have first coordinate triangulate:

so the number of said simplices is and then the number of non-monochromatic simplices for the first choice labeling is . This implies that

as desired. □

Remark 5.1 By (1.2), we have

Hence, the multiplicative constant in the upper bound (1.2) tends to infinity as , whereas the multiplicative constant in the lower bound (1.1) tends to 0 as .

Remark 5.2 For k = 2, q = 1, or q = 2, the lower bounds (1.1) and in Example 4.3 coincide with the upper bound (1.2). Therefore,

6. Conclusions and future directions

In this paper, we address the open problem posed in [6] of determining the minimum number of non-monochromatic simplices in Sperner labelings of triangulations of . In that work, the authors emphasised the importance, for their applications, of obtaining a lower bound for with a tight multiplicative constant.

To this end, we characterise the regular triangulation of and establish the bounds (1.1) and (1.2) for , namely a lower bound of order and an upper bound of order . In addition, for the initial cases k = 2, q = 1, and q = 2, we determine the exact value of . Therefore, for fixed k, our results determine the order of growth of for the regular triangulation and reduce the problem to finding the optimal multiplicative constant.

A natural direction for future work could be to narrow the gap between the multiplicative constant of the lower bound of (Theorem 1.1) and the multiplicative constant in the upper bound (1.2). See Fig 1 for a visualisation of this gap.

In the initial cases for which we have established the exact value of (k = 2, q = 1, q = 2), this value is attained by the first choice labeling. We conjecture that this holds in general.

Our results suggest several research directions for fair division, hypergraph coloring, and combinatorial topology.

Determining the exact constant, or even the lower-order terms, in would sharpen worst-case guarantees for discrete fair-division algorithms. Classical rental-harmony models based on Sperner’s lemma [10], bounded envy-free cake-cutting protocols [2,11,12], and recent studies on mixed-resource division [13] all rely on understanding the structure of worst-case instances. Identifying the true minimum number of non-monochromatic simplices would help characterise these challenging cases and support the design of more robust allocation methods.

Refining the value of could impact rainbow generalisations of the KKM lemma and multi-cut problems such as necklace splitting and the Hobby-rice theorem, where multiple balanced partitions are sought simultaneously. Key developments in this direction include degree-theoretic and combinatorial proofs of Sperner-type results [14–17], polytopal generalisations [18], purely combinatorial approaches [19], rainbow and criticality extensions [20,21], and recent homotopy-based methods [22].

Finally, the regular triangulation considered here is closely related to the edgewise subdivision of a simplex [9] and to earlier work on simplicial mesh generation [8]. A more detailed geometric analysis of these constructions could lead to new algorithmic insights and improved combinatorial bounds.