In this paper we study sum-free subsets of the set {1, . . . , n}, that is, subsets of the first n positive integers which contain no solution to the equation x+ y = z. Cameron and Erdős conjectured in 1990 that the number of such sets is O(2). This conjecture was confirmed by Green and, independently, by Sapozhenko. Here we prove a refined version of their theorem, by showing that the number o...

Let $G$ be a finite simple connected graph on $[n]$ and $R = K[x_1, \ldots, x_n]$ the polynomial ring in $n$ variables over field $K$. The edge ideal of is $I(G)$ $R$ which generated by those monomials $x_ix_j$ for $\{i, j\}$ an $G$. In present paper, possible tuples $(n, {\rm depth} (R/I(G)), reg} \dim R/I(G), deg} \ h(R/I(G)))$, where ${\rm h(R/I(G))$ degree $h$-polynomial $R/I(G)$, arising f...

A poorly known African species Xanthopygoides niger (Cameron, 1951), hitherto a member of a monotypic genus of the subtribe Xanthopygina, is transferred to the subgenus Raucalius Tottenham, 1949 of the genus Philonthus Stephens, 1829, subtribe Philonthina, and redescribed. Th e new genus-group synonymy, Raucalius Tottenham, 1951 = Xanthopygoides Verdcourt, 1952, is established. A new combinatio...

A family of permutations A ⊂ Sn is said to be intersecting if any two permutations in A agree at some point, i.e. for any σ, π ∈ A, there is some i such that σ(i) = π(i). Deza and Frankl [3] showed that for such a family, |A| ≤ (n− 1)!. Cameron and Ku [2] showed that if equality holds then A = {σ ∈ Sn : σ(i) = j} for some i and j. They conjectured a ‘stability’ version of this result, namely th...

Harmonograms are visual designs based on harmonic analyses of strings of characters. Fourier descriptors are synthesized according to character positions and values, and then traditional methods are used to generate the resulting curves. Undersampling introduces visual artifacts that actually enhance the final designs. While their primary use is for the creation of artistic designs, harmonogram...