query: How might I prove that an n-length subset of {1,2,..2n-2} must have at least 2 consecutive members?
positive_passages.docid: 150324
positive_passages.text: Pigeonhole principle: prove that any set of $n+1$ integers from $\{1,2,...,2n\}$ has two consecutive integers differ by one.
negative_passages.docid: 74730
negative_passages.text: Combinatorial Proof of ${{n}\choose{k}} {{k}\choose{j}} = {{n}\choose{j}} {{n-j}\choose{k-j}}$
negative_passages.docid: 129680
negative_passages.text: Prove that $n$ is even.
negative_passages.docid: 5801
negative_passages.text: Combinatorial proof help
negative_passages.docid: 57746
negative_passages.text: Prove that, for any set of consecutive integers with an odd number of terms, the sum of the integers is always a multiple of the number of terms.
negative_passages.docid: 123626
negative_passages.text: Prove that even $n$ can be partitioned to $\frac n2$ edges
negative_passages.docid: 959
negative_passages.text: Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
negative_passages.docid: 83615
negative_passages.text: split conjugacy class in $S_n$ into two equal size conjugacy classes in $A_n$
negative_passages.docid: 70761
negative_passages.text: Number of ways to connect sets of $k$ dots in a perfect $n$-gon
negative_passages.docid: 104855
negative_passages.text: Prove that $N \subset Z(G)$
negative_passages.docid: 4061
negative_passages.text: Proving there are infinitely many pairs of square-full consecutive integers
negative_passages.docid: 22367
negative_passages.text: How do I prove that for every positive integer $n$, there exist $n$ consecutive positive integers, each of which is composite?
negative_passages.docid: 104112
negative_passages.text: Number of finite simple groups of given order is at most $2$ - is a classification-free proof possible?
negative_passages.docid: 123117
negative_passages.text: Is it possible to find $n-1$ consecutive composite integers
negative_passages.docid: 42526
negative_passages.text: The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)
negative_passages.docid: 132605
negative_passages.text: Finding the number of subsets of a set such that an element divides the succeeding element.
negative_passages.docid: 8645
negative_passages.text: A formula for bit strings with no $k$ consecutive $1$'s using generating functions