Frankl-Rödl-type theorems for codes and permutations

Abstract

We give a new proof of the Frankl-Rödl theorem on forbidden intersections, via the probabilistic method of dependent random choice. Our method extends to codes with forbidden distances, where over large alphabets our bound is significantly better than that obtained by Frankl and Rödl. We also apply our bound to a question of Ellis on sets of permutations with forbidden distances, and to establish a weak form of a conjecture of Alon, Shpilka and Umans on sunflowers.

System of unbiased representatives for a collection of bicolorings

“…For r = f (n), where f (n) is an increasing function in n, this establishes only sublinear lower bounds for γ(n, n 2 , r). We use a vector space orthogonality argument combined with a theorem of Keevash and Long [18] to obtain a linear lower bound on γ(n, k, r) under certain restrictions on n, k and r. Theorem 4. Let r = 2c for any odd integer c ∈ {1, .…”

Bisecting and D-secting families for set systems

“…When cn < k < (1 − c)n for a constant c, 0 < c < 1 2 , we establish an improved lower bound for β [±1] (n, k) using a vector space orthogonality argument, enabling us to apply a recent result of Keevash and Long [3].…”

“…For any code C, let d(C) be the set of all the Hamming distances allowed for any x, y ∈ C. A code is called d-avoiding if d d(C). We have the following upper bound on the cardinality of a d-avoiding code C as given in [3].…”

“…Theorem 23 [3] Let C ⊆ [q] n and let ǫ satisfy 0 < ǫ < 1 2 . Suppose that ǫn < d < (1 − ǫ)n and d is even if q = 2.…”

Some old and new problems in combinatorial geometry I: around Borsuk's problem

“…The proof of Frankl-Rödl is an ingenious application (bootstrapping of a kind) of isoperimetric results. Recently Keevash and Long [67] found a new proof of Frankl-Rödl's theorem based on the Frankl-Wilson theorem.…”