A new proof of Delahan's induced-universality result

We give a short and self-contained proof of Delahan’s theorem stating that every simple graph on $n$ vertices occurs as an induced subgraph of a Steinhaus graph on $\frac{n(n-1)}{2}+1$ vertices. This new proof is obtained by considering the notion of generating index sets for Steinhaus triangles.

💡 Research Summary

The paper presents a concise, self‑contained proof of Delahan’s induced‑universality theorem for Steinhaus graphs, which states that every simple graph on n vertices can be realized as an induced subgraph of a Steinhaus graph on tₙ₋₁ + 1 vertices, where tₙ₋₁ = n(n‑1)/2 is the (n‑1)‑st triangular number. The authors achieve this by introducing the notion of generating index sets for Steinhaus triangles and by analyzing certain minors of the infinite binomial matrix.

First, the authors recall the definition of a binary Steinhaus triangle: a down‑pointing triangular array of zeros and ones satisfying the Pascal‑like recurrence a_{i,j} ≡ a_{i‑1,j‑1} + a_{i‑1,j} (mod 2). They note that the entire triangle is uniquely determined by its top row, which yields a vector‑space structure ST(n) of dimension n over the field 𝔽₂.

The central new concept is a “generating index set” A ⊂ T_n of size n such that the projection map π_A : ST(n) → 𝔽₂ⁿ, which extracts the entries indexed by A, is a linear isomorphism. Proposition 2.2 shows that A is generating precisely when the n × n matrix M_A = (C(i_k,ℓ − j_k)) (where C denotes binomial coefficients) is invertible modulo 2. This connects the problem to the Pascal matroid modulo 2, a structure studied in earlier works.

Next, the paper studies determinants of submatrices of the infinite binomial matrix B = (C(i,j)){i,j≥0}. Proposition 3.1 gives a general formula: for any increasing sequence of row indices a₁ < … < a_n, the determinant of the submatrix formed with columns 0,…,n‑1 equals Δ(a₁,…,a_n)·∏{k=0}^{n‑1} k!, where Δ is the Vandermonde product of differences. When the rows are chosen as the first n triangular numbers t₀,…,t_{n‑1}, the Vandermonde product simplifies to ∏_{i=1}^{n‑1} (2i‑1)^{,n‑i}, which is always odd. Consequently, the corresponding minors are invertible modulo 2 (Proposition 3.2).

The authors then define a specific index set A_n = { (t_s, t_{r+1} − t_s − 1) | 0 ≤ s ≤ r ≤ n‑2 }. Lemma 4.2 proves that the matrix M_{A_n} built from this set is block lower‑triangular, with diagonal blocks exactly the binomial minors B