# Increasing sequences with nonzero block sums and increasing paths in edge-ordered graphs

Journal Article (Journal Article)

Consider the maximum length f(k) of a (lexicographically) increasing sequence of vectors in GF(2) with the property that the sum of the vectors in any consecutive subsequence is nonzero modulo 2. We prove that 23 48 · 2 ≤ f(k) ≤ ( 1 2 + o(1))2 . A related problem is the following. Suppose the edges of the complete graph K are labelled by the numbers 1,2,..., ( ). What is the minimum α(n), over all edge labellings, of the maximum length of a simple path with increasing edge labels? We prove that α(n) ≤ ( 1 2 + o(1))n. © 1984. k k k n n 2

### Full Text

### Duke Authors

### Cited Authors

- Calderbank, AR; Chung, FRK; Sturtevant, DG

### Published Date

- January 1, 1984

### Published In

### Volume / Issue

- 50 / C

### Start / End Page

- 15 - 28

### International Standard Serial Number (ISSN)

- 0012-365X

### Digital Object Identifier (DOI)

- 10.1016/0012-365X(84)90031-1

### Citation Source

- Scopus