87 research outputs found
Random algebraic construction of extremal graphs
In this expository paper, we present a motivated construction of large graphs
not containing a given complete bipartite subgraph. The key insight is that the
algebraic constructions yield very non-smooth probability distributions.Comment: 8 page
Non-trivial solutions to a linear equation in integers
For k>=3 let A \subset [1,N] be a set not containing a solution to a_1
x_1+...+a_k x_k=a_1 x_{k+1}+...+a_k x_{2k} in distinct integers. We prove that
there is an epsilon>0 depending on the coefficients of the equation such that
every such A has O(N^{1/2-epsilon}) elements. This answers a question of I.
Ruzsa.Comment: 5 pages, typos corrected, stronger result give
Twins in words and long common subsequences in permutations
A large family of words must contain two words that are similar. We
investigate several problems where the measure of similarity is the length of a
common subsequence.
We construct a family of n^{1/3} permutations on n letters, such that LCS of
any two of them is only cn^{1/3}, improving a construction of Beame, Blais, and
Huynh-Ngoc. We relate the problem of constructing many permutations with small
LCS to the twin word problem of Axenovich, Person and Puzynina. In particular,
we show that every word of length n over a k-letter alphabet contains two
disjoint equal subsequences of length cnk^{-2/3}.
Many problems are left open.Comment: 18+epsilon page
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