# Tagged Questions

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### Can we sometimes define the parity of a set?

Suppose that ${n\choose k}, {n-1\choose k-1}, \ldots, {n-k+1\choose 1}$ are all even. (This happens for example if $k=2^\alpha-1$ and $n=2k$.) In this case, can we select ${n\choose k}/2$ sets of size ...
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### Hitting sets (aka covers aka transversals) of Steiner triple systems

Does there exist a constant $c$ so that the lines of every Steiner triple system on $v$ points can be covered by $cv$ points? That is if $D \in STS(v)$ with point set $T=\{1,2,\ldots,v\}$ then ...
Let $D=(P,L)$ be either a $(v,k,\lambda)$-design or a near-linear space (or, more generally, any incidence structure with "points" and sets of points which are called "blocks" or "lines") and let $S ... 3answers 129 views ### Isomorphism testing in STS(13) What is the simplest isomorphism invariant which can distinguish between the two non-isomorphic Steiner triple systems on$13$points? Train structure and cycle structure, as described here, do the ... 1answer 252 views ### Known results on cyclic difference sets Is there any infinite family of$v$for which all the$(v,k,\lambda)$-cyclic difference sets with$k-\lambda$a prime power coprime to$v$have been determined? A subset$D=\{a_1,\ldots,a_k\}$of ... 1answer 175 views ### a block design question Is it possible to show that every 1-design$D$with$\lambda=4,k=4$on$v$points (for$v$that is a multiple of$3$) contain some 1-design$Q$with$\lambda=1,k=3$on$v$points such that every block ... 1answer 131 views ### Ranks of higher incidence matrices of designs In 1978 Doyen, Hubaut and Vandensavel proved that if$S$is a Steiner triple system$S(2,3,v)$then the$GF(2)$rank of its incidence matrix$N$is $$Rk_{2}(N)=v-(d_{p}+1),$$ where$d_{p}$is the ... 3answers 199 views ### Constructions of$2-(v,3,3)$-designs I am looking for ways to construct an infinite family of designs with parameters$2-(v,3,3)$and apart from some doubling-type recursive constructions (such as in this paper) I haven't found anything ... 0answers 346 views ### Does$(\mathbb{Z}/n\mathbb{Z})^2$ever admit a difference set when$n$is odd? A difference set of a group$G$is a subset$D\subseteq G$with the property that there exists an integer$\lambda>0$such that for every non-identity member$g$of$G$, there exist exactly ... 1answer 77 views ### Lower bounds on cardinality of a union of blocks in a design Let$D$be a$(v,k,\lambda)$-design (repeated blocks are allowed). I would like to get a lower bound on the cardinality of the union of$s$blocks. A naive application of inclusion-exclusion gives ... 2answers 141 views ### Resolvable designs from projective space Resolvable designs are block designs with the additional property that the blocks can be partitioned into partitions of the points. It is easy to see that lines in affine space form a resolvable ... 3answers 364 views ### Orthogonal Latin Square 6*6 I need to make remarks about Tarry's Proof for the nonexistence of 6x6 Latin Squares as part of my final exam for a class I'm in. Problem is, I can't find it ANYWHERE on the internet. I can only find ... 5answers 461 views ### Intersecting 4-sets Is it possible to have more than$N = \binom{\lfloor n/2\rfloor}{2}$subsets of an$n$-set, each of size 4, such that each two of them intersect in 0 or 2 elements? To see that$N$is achievable, ... 1answer 378 views ### Popular elements in cross-intersecting families Let$\mathcal{T}$and$\mathcal{S}$be two families of subsets of$[n]$such that for all$T_i\in \mathcal{T}$and$S_j\in \mathcal{S}$,$|T_i \cap S_j| \neq\emptyset|T_i| , |S_j| \leq t = ...
Question Given a $t-(v,k,\lambda)$ design $(X,\mathcal{B})$ and a set $U\subset X$ with $|U|=u\leq t$, what is the number of blocks $B\in\mathcal{B}$ such that $B\cap U=\emptyset$? The answer is: ...