Forbidding just one intersection for short integer sequences
Elizaveta Iarovikova, Fedor Noskov, Georgy Sokolov, Nikolai Terekhov
TL;DR
The paper advances the Erdős–Sós forbidden intersection problem for words over an alphabet [m] by proving that for m and n polynomial in t, the largest (t−1)-avoiding subfamily of [m]^n has size m^{n−t}, realized only by a t-star when n is sufficiently large. The authors blend spread-approximation techniques with measure-boosting via random gluings and leverage hypercontractivity to control global structure, enabling a decomposition into tau-homogeneous cores and a reduction to incomplete t-stars. They develop two complementary parameter regimes: (i) m and n with poly(t) bounds relative to log m, and (ii) m and n with stronger polynomial dependencies; in both, they deduce the extremal configuration and establish a robust structural description. The results generalize and extend prior work, providing a polynomial-dependency regime where the Erdős–Sós forbidden-intersection phenomenon for codes in [m]^n is resolved, with potential extensions to broader combinatorial objects.
Abstract
In this paper, we study the famous Erdős--Sós forbidden intersection problem for words over alphabet of size $m$: what is the maximal size of a subfamily $\mathcal{F}$ of $[m]^n$ that does not contain two vectors $x, y$ that coincide on exactly $t - 1$ coordinates? We answer this question provided $m \ge poly(t)$, $n \ge poly(t)$ for some polynomial function $poly(\cdot)$ of $t$, greatly extending the recent result of Keevash, Lifshitz, Long and Minzer. Our proof combines some of the recently developed methods in extremal combinatorics, including the spread approximation technique of Kupavskii and Zakharov, and the hypercontractivity approach developed in a series of works of Keevash, Keller, Lifshitz, Long, Marcus and Minzer.