The PFR Conjecture Holds for Two Opposing Special Cases
نویسنده
چکیده
Let A ⊆ F2 be a set with |2A| = K|A|. We prove that if (1) for at least a fraction 1−K−9 of all s ∈ 2A, the set (A+ s)∩A has size at most L · |A|/K, or (2) for at least a fraction K−L of all s ∈ 2A, the set (A+ s)∩A has size at least |A| · (1−K−1/L), then there is a subset B ⊆ A of size |A|/KOL(1) such that span(B) ≤ KOL(1) · |A|.
منابع مشابه
On the oriented perfect path double cover conjecture
An oriented perfect path double cover (OPPDC) of a graph $G$ is a collection of directed paths in the symmetric orientation $G_s$ of $G$ such that each arc of $G_s$ lies in exactly one of the paths and each vertex of $G$ appears just once as a beginning and just once as an end of a path. Maxov{'a} and Ne{v{s}}et{v{r}}il (Discrete Math. 276 (2004) 287-294) conjectured that ...
متن کاملFrankl's Conjecture for a subclass of semimodular lattices
In this paper, we prove Frankl's Conjecture for an upper semimodular lattice $L$ such that $|J(L)setminus A(L)| leq 3$, where $J(L)$ and $A(L)$ are the set of join-irreducible elements and the set of atoms respectively. It is known that the class of planar lattices is contained in the class of dismantlable lattices and the class of dismantlable lattices is contained in the class of lattices ha...
متن کاملNew characterization of some linear groups
There are a few finite groups that are determined up to isomorphism solely by their order, such as $mathbb{Z}_{2}$ or $mathbb{Z}_{15}$. Still other finite groups are determined by their order together with other data, such as the number of elements of each order, the structure of the prime graph, the number of order components, the number of Sylow $p$-subgroups for each prime $p$, etc. In this...
متن کاملA note on Fouquet-Vanherpe’s question and Fulkerson conjecture
The excessive index of a bridgeless cubic graph $G$ is the least integer $k$, such that $G$ can be covered by $k$ perfect matchings. An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless cubic graph has excessive index at most five. Clearly, Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5, so Fouquet and Vanherpe as...
متن کاملAn additive combinatorics approach to the log-rank conjecture in communication complexity
For a {0, 1}-valued matrixM let CC(M) denote the deterministic communication complexity of the boolean function associated with M . The log-rank conjecture of Lovász and Saks [FOCS 1988] states that CC(M) ≤ logc(rank(M)) for some absolute constant c where rank(M) denotes the rank of M over the field of real numbers. We show that CC(M) ≤ c · rank(M)/ log rank(M) for some absolute constant c, ass...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1311.0172 شماره
صفحات -
تاریخ انتشار 2013