2. What is the smallest number, denoted , to guarantee that a -matrix with the entries has 2 neighboring entries such that and ? Obviously, , where is the smallest number such that any subset of of size has at least neighbors. Due to Theorem 11, is the smallest number such that the initial segment of length in the simplicial order on has neighbors. In your proof you used the fact that , which implies by Theorem 11 that . However this bound is not tight. Consider the hamming ball . It is easy to see that and . Thus . What about higher dimensions?

## Related questions: continued

2. What is the smallest number, denoted , to guarantee that a -matrix with the entries has 2 neighboring entries such that and ? Obviously, , where is the smallest number such that any subset of of size has at least neighbors. Due to Theorem 11, is the smallest number such that the initial segment of length in the simplicial order on has neighbors. In your proof you used the fact that , which implies by Theorem 11 that . However this bound is not tight. Consider the hamming ball . It is easy to see that and . Thus . What about higher dimensions?