Updated information on "Positivity problems and
conjectures in algebraic combinatorics"
 A solution to
Problem 2 for convex polytopes was given by Kalle
Karu, with a strengthening
by Paul Bressler and Valery A. Lunts.
 Some progress
on Problem 4 is due to Victor Reiner and his
colloborators Naichung Conan Leung, Dennis Stanton, and Volkmar
Welker. The paper with Welker is also related to Problem 20.
 The conjecture following Problem 9 is false as stated. Pavlo
Pylyavskyy pointed out that if λ ≥ λ' then it may not
be the case that Π_{λ} ≤
Π_{λ'}, the smallest example
being λ = (6,2,2,1,1) and λ' = (5,3,1,1,1,1). The
conjecture remains open if modified to state that
φ_{λ} has
full rank when λ ≥ λ', though this may very
well be false. The original conjecture remains open in the case that
would imply the Foulkes plethysm conjecture, viz., λ has
m parts equal to n, where n ≥
m. However, a counterexample in that case was announced by
J. Müller and Max Neunhoeffer, but details have not yet
appeared. Update: see the addendum in the paper
"Some computations regarding Foulkes' conjecture" by J. Müller
and Max Neunhoeffer, Experimental
Mathematics 14 (2005), No. 3, 277283.
 A solution to the conjecture of Sundaram mentioned shortly after
Problem 9 was found by Ben Joseph in April, 2000.
 A solution
to Problem 13 (the nonnegativity of the coefficients
of the (q,t)Kostka polynomials) has been given by Mark
Haiman.
 Problem 23 was solved
affirmatively by Maria Chudnowsky and Paul Seymour in 2004.
 Problem 5 has a negative answer. I don't remember the
reference.

A negative
answer to Problem 6 was found by Art Duval, Caroline Klivans,
and Jeremy Martin.
 A positive
answer to Problem 7 was found by Kalle Karu.

A negative
answer was found by Petter Brändén for general
(i.e., not necessarily natural)
labelled posets. Subsequently John Stembridge gave a
negative
answer for naturally labelled posets.
 Problem 24 was given a positive answer by
Petter Brändén, James Haglund, Mirkó
Visontai, and David Wagner.
 Problem 25: update forthcoming.