Aha! Your neighbor has just created the perfect environment for those little oak suckers to emerge. This is a favorite question asked by Central Texas homeowners. I like the answer that the A&M Horticulture website had about this.
I quote in part: “LEARN TO LOVE THEM because the more you cut the more they multiply. Some homeowners keep them trimmed at a certain height and interplant ground cover -- the ground cover conceals the oak sprout since the foliage is similar. We DO NOT recommend a herbicide application to sprouts since the sprouts are connected directly to the Mother tree and could cause damage. Roundup herbicide will do little more than defoliate the sprouts and could damage the tree to which they are attached. Learn to co-exist; they can become one of the most drought-tolerant ground covers available.”
Multiple previous answers by Mr Smarty Plants tend to pretty much agree with this; I will reference those answers in the summary that follows.
They lean towards cutting the suckers regularly, and even recommend sharpshooters and garden loppers as the tools of choice. Be careful though, as this exposes the tree to Oak Wilt, so trimming these between February and June is highly discouraged.
From my own experience, I can echo the advice given above and add to it a little bit. We have two lovely Live Oaks in our front yard. We opened the area under them when we converted the yard to a native garden. Those suckers immediately started coming up and we had a large area where they were pretty thick. I would mow and weed-whack them for normal control, then once or twice a year I’d dig as much as possible and cut some of the larger roots with loppers or shears. Later I graduated to using a hedge trimmer under the soil [REPEAT: Do not cut the Oak from February to June]
Lately, we have converted from this to a covered situation similar to that recommended in this comment. We laid down multiple layers of newspaper, covered by heavy cardboard, covered by 4-6” of mulch. Using cardboard and paper allows some water to seep through, unlike the pond cover suggestion. You should cover everything up to maybe 6” from the tree trunk. We are now about 6 months into this and none have made it up yet. I’m very hopeful, keep your fingers crossed for us. Still, as they said in Jurassic Park: “Life will find a way”.