Cristian Budala | Searcharxiv

arXiv2026

Sunflower-Free Uniform Families: Recursive Constructions and Explicit Bounds

Let $f(w,k)$ be the maximum size of a $w$-uniform family containing no sunflower with $k$ petals. We introduce a recursive construction for sunflower-free families and use it to obtain a general lower bound on the exponential growth rate of $f(w,k)$. We also prove a general upper bound for $3$-uniform families with at least four petals. Our results give $39\le f(3,4)\le49$, $f(3,5)\le146$, $153\le f(3,6)\le255$, $259\le f(3,7)\le474$, and $54\le f(4,3)\le83$. In addition, we prove that the maximum size of an intersecting $4$-uniform family containing no sunflower with three petals is $27$. The upper bounds $49$ and $83$ are computer-assisted. The finite lower bounds come from explicit constructions.

math.CO↗

arXiv2026

An Exhaustive Census of Main-Diagonal Symmetric Costas Arrays of Orders 37-42

We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over $\mathbb{F}_{41}$ or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over $\mathbb{F}_{43}$; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, $n^2/18+n^{3/2}/360$, with an $O(n)$ remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.

math.CO↗