![](/files/happy5.png)
zero sum
Pebbling a cartesian product ★★★
Author(s): Graham
We let denote the pebbling number of a graph
.
Conjecture
.
![$ p(G_1 \Box G_2) \le p(G_1) p(G_2) $](/files/tex/3371d3b2280f593cc1fc3a0b3da7a685c3525910.png)
Davenport's constant ★★★
Author(s):
For a finite (additive) abelian group , the Davenport constant of
, denoted
, is the smallest integer
so that every sequence of elements of
with length
has a nontrivial subsequence which sums to zero.
Conjecture
![$ s( {\mathbb Z}_n^d) = d(n-1) + 1 $](/files/tex/2bea401874dc0a9cd37e10b6df927b12a7ce2402.png)
Keywords: Davenport constant; subsequence sum; zero sum
Bases of many weights ★★★
Let be an (additive) abelian group, and for every
let
.
Conjecture Let
be a matroid on
, let
be a map, put
and
. Then
![$ M $](/files/tex/3f02401f624e31ef8679d3c3628c1f310058f388.png)
![$ E $](/files/tex/aedbef97f3db174b677f00be580a095e7fefa310.png)
![$ w : E \rightarrow G $](/files/tex/7c1a9b6ba2a67b002e25339803f3d5a6da1b684d.png)
![$ S = \{ \sum_{b \in B} w(b) : B \mbox{ is a base} \} $](/files/tex/15b6767566360480b09b1982f902c978265de9c9.png)
![$ H = {\mathit stab}(S) $](/files/tex/e338cf1ff9295e47c3f1552771eb4fecfb4d730f.png)
![$$|S| \ge |H| \left( 1 - rk(M) + \sum_{Q \in G/H} rk(w^{-1}(Q)) \right).$$](/files/tex/e22b2f4b60bf6bbfd304dd219b3092fb50cbae67.png)
Gao's theorem for nonabelian groups ★★
Author(s): DeVos
For every finite multiplicative group , let
(
) denote the smallest integer
so that every sequence of
elements of
has a subsequence of length
(length
) which has product equal to 1 in some order.
Conjecture
for every finite group
.
![$ s'(G) = s(G) + |G| - 1 $](/files/tex/91264d9b321ddb6a08656e574ccf314d9ebfa121.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: subsequence sum; zero sum
Few subsequence sums in Z_n x Z_n ★★
Conjecture For every
, the sequence in
consisting of
copes of
and
copies of
has the fewest number of distinct subsequence sums over all zero-free sequences from
of length
.
![$ 0 \le t \le n-1 $](/files/tex/3f439dde8bba9a34c4b73a7bf35d2ba2d600dd53.png)
![$ {\mathbb Z}_n^2 $](/files/tex/784102d74f41429c112d0dd6746a4ab9f1957afe.png)
![$ n-1 $](/files/tex/da6174078cbeae6601684c08526200d9254caa11.png)
![$ (1,0) $](/files/tex/02e6ed02ec9ede67b905b1ca3c64be3eb3c6f11b.png)
![$ t $](/files/tex/4761b031c89840e8cd2cda5b53fbc90c308530f3.png)
![$ (0,1) $](/files/tex/2f2f87361c58fc118cefb1ab5cb288a25e20007f.png)
![$ {\mathbb Z}_n^2 $](/files/tex/784102d74f41429c112d0dd6746a4ab9f1957afe.png)
![$ n-1+t $](/files/tex/26b23f9cc4119b690ed97ea8d21da62ec7899f64.png)
Keywords: subsequence sum; zero sum
Olson's Conjecture ★★
Author(s): Olson
Conjecture If
is a sequence of elements from a multiplicative group of order
, then there exist
so that
.
![$ a_1,a_2,\ldots,a_{2n-1} $](/files/tex/fb729d891bb896d89e505b8df270836848cbe79a.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ 1 \le j_1 < j_2 \ldots < j_n \le 2n-1 $](/files/tex/d419cf48afd869605409388bdf916991759fb865.png)
![$ \prod_{i=1}^n a_{j_i} = 1 $](/files/tex/594ddd47f49bdb50d25a80ddde1173c0c50ccf75.png)
Keywords: zero sum
![Syndicate content Syndicate content](/misc/feed.png)