## Section: New Results

### Algorithms and combinatorics

Participant : Laurent Alonso.

#### Bounds for Cops and Robber Pursuit

We prove that the robber can evade (that is, stay at least unit distance from) at least cops patroling an n×n continuous square region, that a robber can always evade a single cop patroling a square with side length 4 or larger, and that a single cop on patrol can always capture the robber in a square with side length smaller than (with E.M. Reingold, submitted to J. of Computational Geometry Theory and Applications).

#### Average-Case Lower Bounds for the Plurality Problem

Given a set of n elements, each of which is colored one of c2
colors, we have to determine an element of the plurality (most frequently
occurring) color by pairwise equal/unequal color comparisons of elements. We
derive lower bounds for the expected number of color comparisons when the
c^{n} colorings are equally probable. We prove a general lower bound of
for c2 ; we prove the stronger particular
bounds of for c = 3 ,
for c = 4 , for c = 5 , for c = 6 , for c = 7 , and
for c = 8 (with E.M. Reigold, [11] ).