Abstract
Let \([n]=\{1,2,\dots , n\}\). For each \(i\in [k]\) and \(j\in [n]\), let \(w_{i}(j)\) be a real number. Suppose that \(\sum _{i\in [k],j\in [n]} w_{i}(j)\ge 0\). Let \(\mathcal F\) be the set of all functions with domain [k] and codomain [n]. For each \(f\in \mathcal F\), let
A function \(f\in \mathcal F\) is said to be nonnegative if \(w(f)\ge 0\). Let \(\mathcal F^+(w)\) be set of all nonnegative functions, i.e.,
In this paper, we show that \(\vert \mathcal F^+(w)\vert \ge n^{k-1}\) for \(n\ge 3(k-1)^2\).
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1 Introduction
Let \([n]=\{1,2,\dots , n\}\). Suppose that each \(i\in [n]\) is assigned a weight \(w_i\) such that \(\sum _{i=1}^n w_i\ge 0\). For each \(S\subseteq [n]\), let \(w(S)=\sum _{s\in S} w_s\). Let
Theorem 1.1
If \(n\ge 4k\), then
for all w.
The above theorem was first conjectured by Manickam, Miklós and Singhi [8, 9]. Chowdhury [3] showed that Theorem 1.1 is true for \(k=3\) and with an improved bound of \(n\ge 11\). Marino and Chiaselotti [10] showed that the theorem is true for \(k\le 3\) and Manickam and Singhi [9] showed that the theorem is true when \(n\equiv 0 \mod k\). Some authors are able to show that Eq. (1.1) holds if n is large compared to k. In fact, Manickam and Miklós [8] showed that Eq. (1.1) holds if \(n\ge (k-1)(k^k+k^2) +k\). Tyomkyn [12] improved the bound to \(n\ge k(4e \ln k)^k\). The first polynomial bound \(n\ge 33k^2\) was obtained by Alon et al. [1]. The bound was improved to \(n\ge 8k^2\) by Chowdhury et al. [4]. Frankl [5] gave a short proof with a bound of \(n\ge 3k^3/2\). A linear bound \(n\ge 10^{46}k\) was obtained by Pokrovskiy [11]. Finally, the theorem was proved by Blinovsky [2].
An analogue of Theorem 1.1 for vector spaces has been proved by Chowdhury et al. [4], Huang and Sudakov [6] and Ihringer [7]. In this paper, we will consider a version of the problem for certain function.
For each \(i\in [k]\) and \(j\in [n]\), let \(w_{i}(j)\) be a real number. The w is called a weight assignment for ([k], [n]). Suppose that the weight assignment satisfies \(\sum _{i\in [k],j\in [n]} w_{i}(j)\ge 0\). Let \(\mathcal F\) be the set of all functions with domain [k] and codomain [n]. For each \(f\in \mathcal F\), let
A function \(f\in \mathcal F\) is said to be nonnegative if \(w(f)\ge 0\). Let \(\mathcal F^+(w)\) be set of all nonnegative functions, i.e.,
Theorem 1.2
If \(n\ge 3(k-1)^2\), then
for all weight assignment w for ([k], [n]) satisfying \(\sum _{i\in [k],j\in [n]} w_{i}(j)\ge 0\). Moreover, equality holds if and only if there is a \(i_0\in [k]\) and \(j_0\in [n]\) such that
It is noticed that by the result of Huang and Sudakov [6], we can deduce the lower bound \(n^{k-1}\). In fact, let \(\mathcal {H}\) be the hypergraph with the vertex set V and the edge set E given as follows:
Then the number of nonnegative edges is at least \(\delta (H) = n^{k-1}\). However, the result for hypergraphs holds only for \(n \ge 10k^{3}\) which is larger than the present bound \(3(k-1)^{2}\).
2 Proof of Theorem 1.2
For each \(i\in [k]\), let \(\{a_{i1},a_{i2},\dots , a_{in}\}=[n]\) be arranged such that
Case 1 Suppose that there is an \(i_0\in [k]\) such that
Let \(\mathcal H_0=\{ f\in \mathcal F \ : \ f(i_0)=a_{i_01}\}\). For each \(g\in \mathcal H_0\), \(w_{i_0}(g(i_0))=w_{i_0} (a_{i_01})\) and \(w_i(g(i))\ge w_i(a_{in})\) for all \(i\in [k], i\ne i_0\) Therefore,
Hence, \(\mathcal H_0\subseteq \mathcal F^+(w)\), and equality holds if and only if \(\mathcal H_0= \mathcal F^+(w)\), because \(\vert \mathcal H_0\vert =n^{k-1}\).
Case 2 We may assume that for all \(i'\in [k]\)
We will show that \(\vert \mathcal F^+(w)\vert >n^{k-1}\).
Let
Case 2.1 Suppose \(T\ge 0\).
Assume at the moment that
We will show that \(T<0\), and this contradicts our assumption. Let the column vectors \(\mathbf b_1,\mathbf b_2,\dots , \mathbf b_k\in \mathbb R^k\) be defined as
For each \(i,j\in [k]\), let \(\mathbf b_{i}(j)\) be the element in the jth row of \(\mathbf b_i\). Note that \(\mathbf b_{i}(i)=w_i(a_{i1})\) and if \(j\ne i\), then
Let
Note that B is a \(k\times k\) matrix and
Now consider the ith row sum of the matrix B. Note that
where the last inequality follows from equation (2.2). By summing up all the row in B, we have \(T=\sum _{1\le i,j\le k} c_{ij}<0\), a contradiction.
Thus, there is an \(i_0\in [k]\) such that
We will show that there is another \(i_1\in [k]\setminus \{i_0\}\) with
Assume at the moment that
We will show that \(T<0\), and this contradicts our assumption.
If \(i_0=k\), then let \(\overline{B}=B\), and if \(i_0\ne k\), then let
where \(\overline{\mathbf b}_i=\mathbf b_i\) for \(i\notin \{i_0,k\}\),
For each \(i,j\in [k]\), let \(\overline{\mathbf b}_{i}(j)\) be the element in the jth row of \(\overline{\mathbf b}_i\). Note that \(\overline{\mathbf b}_{i_0}(k)=w_{i_0}(a_{i_01})\) and if \(j\ne k\), then
Furthermore, \(\overline{\mathbf b}_{k}(i_0)=w_{k}(a_{k1})\) and if \(j\ne i_0\), then
By Eq. (2.1), the kth row sum of the matrix \(\overline{B}\) is
If \(i\notin \{i_0, k\}\), then the ith row sum of the matrix \(\overline{B}\) is
Finally, the \(i_0\)-th row sum of the matrix \(\overline{B}\) is
Therefore, \(T=\sum _{1\le i,j\le k} \overline{c}_{ij}<0\), a contradiction.
Hence, Eqs. (2.4) and (2.5) hold. For each \(i\in [k]\), let \(U_i=\{ a_{i2}, a_{i3},\dots , a_{i(n-k+2)}\}\) and
Note that \(\vert \mathcal H_1\vert =\vert \mathcal H_2\vert =(n-k+1)^{k-1}\). Let \(f\in \mathcal H_1\). Then, by Eq. (2.4),
So, \(f\in \mathcal F^+(w)\) and \(\mathcal H_1\subseteq \mathcal F^+(w)\). Similarly, by using Eq. (2.5), \(\mathcal H_2\subseteq \mathcal F^+(w)\). Note that \(\mathcal H_1\cap \mathcal H_2=\varnothing \). Therefore,
Hence, \(\vert \mathcal F^+(w)\vert >n^{k-1}\) provided that \(n\ge 3(k-1)^2\).
Case 2.2 Suppose \(T< 0\).
We will show that \(\sum _{i\in [k], n-k+2\le j\le n} w_i(a_{ij})<0\). Assume at the moment that
Then, there is a \(n-k+2\le j_0\le n\) such that \(\sum _{i\in [k]} w_i(a_{ij_0})\ge 0\). This implies that
Thus, \(T=\sum _{i=1}^k w_i(a_{i1})+\sum _{i\in [k], n-k+2\le j\le n} w_i(a_{ij})\ge 0\), a contradiction. Hence,
Since \(\sum _{i\in [k],j\in [n]} w_{i}(j)\ge 0\),
Let \(V_i=\{ a_{i1}, a_{i2},\dots , a_{i(n-k+1)}\}\) and
Let
Note that each \(f\in \mathcal H\) can be represented in the following form
where \(f(i)=b_i\) for all i. This is an one-to-one correspondence between \(\mathcal H\) and \(\mathcal B\).
A subset \(\{f_1,f_2,\dots ,f_{n-k+1}\}\subseteq \mathcal H\) is said to be a partition of \(\mathcal B\) if
Claim 1If\(\{f_1,f_2,\dots ,f_{n-k+1}\}\)is a partition of\(\mathcal B\), then there exist two integers\(j_0, j_1\in [n-k+1]\)with\(w(f_{j_0}), w(f_{j_1})> 0\).
Proof
Note that
where the last inequality follows from Eq. (2.7). Therefore, there is an \(j_0\in [n-k+1]\) with \(w(f_{j_0})>0\). Next,
Since
it follows that
Hence, there is an \(j_1\in [n-k+1]\setminus \{j_0\}\) with \(w(f_{j_1})>0\). This completes the proof of Claim 1. \(\square \)
The number of partitions of \(\mathcal B\) is
Let \(\mathcal A_1, \mathcal A_2, \dots , \mathcal A_l\) be all the partitions of \(\mathcal B\). Note that each \(f\in \mathcal H\) is contained in exactly \(((n-k)!)^{k-1}\) number of \(\mathcal A_i\). Let \({\mathcal {H}}\cap {F}^+(w)=\{g_1,g_2,\dots , {g}_{s}\}\). Since each \(\mathcal A_i\) contains at least 2 \(g_j\)’s (Claim 1) and each \(g_j\) is contained in exactly \(((n-k)!)^{k-1}\) number of \(\mathcal A_i\), we conclude that
So,
provided that \(n\ge 3(k-1)^2\). This completes the proof of Theorem 1.2. \(\square \)
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We would like to thank the anonymous referees for the comments that helped us make several improvements to this paper.
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Communicated by Xueliang Li.
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Ku, C.Y., Wong, K.B. On the Number of Nonnegative Sums for Certain Function. Bull. Malays. Math. Sci. Soc. 43, 15–24 (2020). https://doi.org/10.1007/s40840-018-0661-6
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DOI: https://doi.org/10.1007/s40840-018-0661-6